Analysis and Optimal Control of Propagation Model for Malware in Multi-Cloud Environments with Impact of Brownian Motion Process

Abstract

Today, cloud computing is a widely used technology that provides a wide range of services to numerous sectors around the world. This technology depends on the interaction and cooperation of virtual machines (VMs) to complete various computing tasks, propagating malware attacks quickly due to the complexity of cloud computing environments and users’ interfaces. As a result of the rising demand for cloud computing from multiple perspectives for complete analysis and decision-making across a range of life disciplines, multi-cloud environments (MCEs) are established. Therefore, in this work, we discuss impacted mathematical modeling for the MCEs’ network dynamics using two deterministic and stochastic approaches. In both approaches, appropriate assumptions are considered. Then, the proposed networks’ VMs are classified to have six different possible states covering media, healthcare, finance, and educational servers. After that, the two developed modeling approaches’ solution existence, uniqueness, equilibrium, and stability are carefully investigated. Using an optimal control strategy, both proposed models are tested for sustaining a certain level of security of the VMs’ states and reducing the propagation of malware within the networks. Finally, we verify the theoretical results by employing numerical simulations to track the malware’s propagation immunization. Results showed how the implemented control methods maintained the essential objectives of managing malware infections.

1. Introduction

In cloud computing, virtualization is a crucial method that enables the transcendence of temporal and physical boundaries. It accomplishes this by splitting a single physical computing resource into several virtual machines (VMs) with identical functions. This enables VM migration [1] to enable the on-demand deployment of computing resources. Regretfully, malware attacks are becoming more frequent targets for the new hidden risks that virtualization has brought about [2]. In addition to resulting in financial losses, these vulnerabilities have the potential to seriously injure people and organizations and even endanger lives [3]. Thus, investigating practical ways to protect virtual environments against malware attacks in the cloud is essential. The issue of malware spreading in cloud systems has grown in prominence due to the rapid development of the cloud computing industry [4]. Large-scale, quickly spreading, and hard-to-track-and-control malware are typical traits found in cloud systems. Furthermore, cloud providers enable users to install applications and upload items (such pictures, movies, and other documents) to the cloud. This makes it possible for hackers to infiltrate the cloud with malware like ransomware and spyware. In addition, ransomware attacks rank among the most prevalent in recent years, indicating that hackers are increasingly likely to exploit user private data for financial benefit [5]. For this purpose, numerous related efforts have been presented by researchers to control the spread of malware in cloud systems. These efforts mostly consist of malware detection, analysis of the malware propagation path, and control of the malware propagation. In cloud systems, malware detection is an essential technique for limiting the spread of malware. To detect malware, researchers currently use feature-based and machine learning-based approaches [6]. Researchers can successfully identify malware by examining its behavior and code [7]. Understanding the laws and mechanisms of malware propagation in cloud systems also requires an understanding of malware propagation paths. Furthermore, one of the most important technologies for limiting the spread of malware in cloud systems is malware propagation control. Certain models of the epidemic dynamics of malware spread have been proposed, based on comparisons between malware and biological counterparts. These models include the SEIQR model [8], the SIR1R2 model [9], the SE1E2IQR model [10], the SIWQ model [11], the SIAR model [12,13], the SEIRF model [14], the SDIQTPR model [15], and the SV1V2EIRF model [16]. These models, however, are not specifically made for cloud environments. The authors in [17] suggested a five-state dynamic model—susceptible, infected high, infected low, first recovered, and completely recovered—to enhance the security and resilience of IoT networks against malicious activities. This model offers a robust solution for mitigating cyber threats using model predictive control (MPC) to effectively manage malware propagation rates by reducing the number of infected IoT devices during periods of high infection. In [18], the authors applied a different optimal control strategy in the dynamic model in [17], considering the count of IoT devices infected during the high infection stage, costs related to executing the immediate response strategy for the restricted environment and additional security defenses, and the costs linked to the installation of the latest and most resilient cybersecurity patch during the complete recovery stage. The SPI (susceptible–protected–infected) model was presented by the authors of [19], who investigated the impact of anti-malware with infectious nodes in cloud environments. However, they made the assumption that all machines interacting with the cloud were vulnerable, which is not realistic in real-world scenarios, because the cost and efficacy of malware control are beyond the scope of this model. Moreover, the authors assert that this is the first mathematical model to investigate how anti-malware software affects malware’s ability to proliferate in cloud environments. Conversely, the susceptible–infected–protected–susceptible (SIPS) model was a novel dynamical model put forth by the authors in [20]. Nonetheless, it is important to note that this model assumes that all virtual machines (VMs) join the virtual network cleanly and were developed in an initially uninfected cloud environment.

In light of all potential devices joining the presumptive dynamic cloud network, this work introduces novel six-state VMs, deterministic and stochastic dynamic models, and four popular multi-cloud environments: media, healthcare, finance, and educational servers with solution existence, uniqueness, positiveness, and stability examination. In addition, a unique integration of assumptions in the cloud network simulates the reality of defensive behavior against malware propagation. Further, the dynamic interaction between the six-state VMs and malware propagation is fully studied and visualized. Finally, a security control function is employed to control the dynamic behavior of some states for investment loss reduction against malware propagation.

The remaining material of our work is organized as follows: Section 2 introduces the suggested deterministic dynamic model of multi-cloud environments with mathematical uniqueness and existence investigation and presents the model optimality control criteria for minimizing a predefined cost function of servers’ isolation, tracing, and protection. Section 3 investigates the constructed stochastic model of multi-cloud environments with full mathematical analysis and shows how model controllability can be achieved. Finally, Section 4 shows the two obtained models numerical results and conclusions.

2. Deterministic MMCE Model

With reference to the flowchart presented in Figure 1, we accumulate a mathematical model of malware in multi-cloud environments (MMCE) that consists of twelve ordinary differential equations. The compartments of the mathematical model are divided into susceptible servers $S\left(\mathfrak{Z}\right)$, infected academic and educational servers $I_1\left(\mathfrak{Z}\right)$, infected healthcare servers $I_2\left(\mathfrak{Z}\right)$, infected financial services servers $I_3\left(\mathfrak{Z}\right)$, infected media servers $I_4\left(\mathfrak{Z}\right)$, isolated academic and educational servers $Q_1\left(\mathfrak{Z}\right)$, isolated healthcare servers $Q_2\left(\mathfrak{Z}\right)$, isolated financial services servers $Q_3\left(\mathfrak{Z}\right)$, isolated media servers $Q_4\left(\mathfrak{Z}\right)$, traced servers $T\left(\mathfrak{Z}\right)$, semi-protected servers $P_1\left(\mathfrak{Z}\right)$, and protected servers $P_2\left(\mathfrak{Z}\right)$, where $S\left(\mathfrak{Z}\right)+{\sum }_{k=1}^4{I}_k\left(\mathfrak{Z}\right)+{\sum }_{k=1}^4{Q}_k\left(\mathfrak{Z}\right)+T\left(\mathfrak{Z}\right)+{\sum }_{k=1}^2{P}_k\left(\mathfrak{Z}\right)=N\left(t\right)$.

Through Figure 1, the following presumptions are taken into account:

(1)

All of the involved transmission rates are positive.

(2)

New servers are added to the dynamic network in every possible network state.

(3)

The virus can only propagate across infected servers.

(4)

Susceptible servers have the ability to transition from one state to the protected one without going through further phases.

(5)

Without first going through the tracing state, isolated servers are unable to transition to the semi-protected state.

(6)

Traced servers have the ability to enter the protective state instantly.

(7)

There is a constant rate of disconnection for every server on the network.

We are now on the model formulation. The flow diagram presented in Figure 1 illustrates how the various states of the servers nodes display behaviors that may be explained by the subsequent governing equations.

System (1) presents the suggested deterministic model equations:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathbf{d}S\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_3+{\xi }_{2}S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_4+{\xi }_{3}S\left(\mathfrak{Z}\right)I_3\left(\mathfrak{Z}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_5+{\xi }_{4}S\left(\mathfrak{Z}\right)I_4\left(\mathfrak{Z}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_6+g_1{I}_1\left(\mathfrak{Z}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_7+g_2{I}_2\left(\mathfrak{Z}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_8+g_3{I}_3\left(\mathfrak{Z}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_9+g_4{I}_4\left(\mathfrak{Z}\right)-({\Theta }_4+d)Q_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}T\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{Z}\right)+{\Theta }_2{Q}_2\left(\mathfrak{Z}\right)+{\Theta }_3{Q}_3\left(\mathfrak{Z}\right)+{\Theta }_4{Q}_4\left(\mathfrak{Z}\right)-(a_1+a_2+d)T\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{11}+a_{1}T\left(\mathfrak{Z}\right)-(\Delta +d)P_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+\Delta P_1\left(\mathfrak{Z}\right)+\alpha S\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right)\hfill \end{array}$$

(1)

with initial conditions

$$\begin{array}{ccc}& & S\left(0\right)=\breve{S}, I_1\left(0\right)=\breve{I_1},\cdots , I_4\left(0\right)=\breve{I_4}, Q_1\left(0\right)=\breve{Q_1},\cdots , Q_4\left(0\right)=\breve{Q_4}, \hfill \\ & & T\left(0\right)=\breve{T}, P_1\left(0\right)=\breve{P_1}, P_2\left(0\right)=\breve{P_2}\hfill \end{array}$$

(2)

where the feasible region is

$$\begin{array}{ccc}\hfill \Xi & =& \{(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\in {\mathbb{R}}_{12}^{+}:0\le S+\sum _{k=1}^4{I}_k+\sum _{k=1}^4{Q}_k+T+P_1+P_2\hfill \\ & \le & \sum _{k=1}^{12}{\eta }_k/d\}\hfill \end{array}$$

Table 1. Dynamic network transmission rates.

Rates	Description
${\eta }_1$	Entering Rate of Susceptible Servers
${\eta }_2$	Entering Rate of Infected Academic and Educational Servers
${\eta }_3$	Entering Rate of Infected Healthcare Servers
${\eta }_4$	Entering Rate of Infected Financial Services Servers
${\eta }_5$	Entering Rate of Infected Media Servers
${\eta }_6$	Entering Rate of Isolated Academic and Educational Servers
${\eta }_7$	Entering Rate of Isolated Healthcare Servers
${\eta }_8$	Entering Rate of Isolated Financial Services Servers
${\eta }_9$	Entering Rate of Isolated Media Servers
${\eta }_{10}$	Entering Rate of Traced Servers
${\eta }_{11}$	Entering Rate of Semi-protected Servers
${\eta }_{12}$	Entering Rate of Protected Servers
${\xi }_1$	Infection Rate of Academic and Educational Servers
${\xi }_2$	Infection Rate of Healthcare Servers
${\xi }_3$	Infection Rate of Financial Services Servers
${\xi }_4$	Infection Rate of Media Servers
${\delta }_1$	Reinstalling System Rate of Infected Academic and Educational Servers
${\delta }_2$	Reinstalling Rate of Healthcare Servers
${\delta }_3$	Reinstalling Rate of Financial Services Servers
${\delta }_4$	Reinstalling Rate of Media Servers
$g_1$	Isolation Rate of Infected Academic and Educational Servers
$g_2$	Isolation Rate of Infected Healthcare Servers
$g_3$	Isolation Rate of Infected Financial Services Servers
$g_4$	Isolation Rate of Infected Media Servers
${\Theta }_1$	Tracing Rate of Isolated Academic and Educational Servers
${\Theta }_2$	Tracing Rate of Isolated Healthcare Servers
${\Theta }_3$	Tracing Rate of Isolated Financial Services Servers
${\Theta }_4$	Tracing Rate of Isolated Media Servers
$a_1$	Transmission Rate from Traced into Semi-protected Servers
$a_2$	Transmission Rate from Traced into Protected Servers
$\Delta$	Transmission Rate from Semi-protected into Protected Servers
$\mu$	Antivirus Expiry Rate
$\alpha$	Antivirus Installing Rate
d	Removal Rate of the Cloud Environment
$\kappa$	Stochastic Diffusion Coefficient

provides a comprehensive interpretation of the parameters used in the model.

Lemma 1.

The set Ξ is positive invariant and attracts all solutions in ${\mathbb{R}}_{12}^{+}$.

Remark 1.

Let the solution set

$$\mathfrak{S}\left(\mathfrak{Z}\right)\equiv \left(S\left(\mathfrak{Z}\right), I_1\left(\mathfrak{Z}\right),\cdots , I_4\left(\mathfrak{Z}\right), Q_1\left(\mathfrak{Z}\right),\cdots , Q_4\left(\mathfrak{Z}\right), T\left(\mathfrak{Z}\right), P_1\left(\mathfrak{Z}\right), P_2\left(\mathfrak{Z}\right)\right)$$

Lemma 2.

Let the initial data set be $\mathfrak{S}\left(\mathfrak{Z}\right){\mid }_{\mathfrak{Z}=0}\ge 0\in \Xi$; then, the solution set $\mathfrak{S}\left(\mathfrak{Z}\right)$ of the model (1) is non-negative ∀$\mathfrak{Z}>0$.

Remark 2.

Since $S+{\sum }_{k=1}^4{I}_k+{\sum }_{k=1}^4{Q}_k+T+P_1+P_2=N$, then all functions in model (1) are bounded,

$$\begin{array}{c}\hfill \begin{array}{c}0\le S\left(\mathfrak{Z}\right)\le N, 0\le I_1\left(\mathfrak{Z}\right)\le N,\cdots , 0\le I_4\left(\mathfrak{Z}\right)\le N, 0\le Q_1\left(\mathfrak{Z}\right)\le N,\cdots , 0\le Q_4\left(\mathfrak{Z}\right)\le N\\ 0\le T\left(\mathfrak{Z}\right)\le N, 0\le P_1\left(\mathfrak{Z}\right)\le N, 0\le P_2\left(\mathfrak{Z}\right)\le N.\end{array}\end{array}$$

2.1. Existence and Uniqueness

The existence and uniqueness of the deterministic MMCE model solutions ((1) and (2)) are examined in this section. We apply the well-known Banach fixed-point theorem to demonstrate the existence of the solution to model one described in (3). To prove the existence and uniqueness of the solution, we execute the following: employing the integral operator ${\int }_0^{\mathfrak{Z}}(·)d\mathfrak{Z}$ on the equations (Eqns.) in model (1) yields

$$\begin{array}{ccc}\hfill S\left(\mathfrak{Z}\right)& =& \breve{S}+{\int }_0^{\mathfrak{Z}}({\eta }_1-{\xi }_1{I}_1\left(\mathfrak{G}\right)S\left(\mathfrak{G}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{G}\right)-{\xi }_3{I}_3\left(\mathfrak{G}\right)S\left(\mathfrak{G}\right)-{\xi }_4{I}_4\left(\mathfrak{G}\right)S\left(\mathfrak{G}\right)\hfill \\ & & +{\delta }_1{I}_1\left(\mathfrak{G}\right)+{\delta }_2{I}_2\left(\mathfrak{G}\right)+{\delta }_3{I}_3\left(\mathfrak{G}\right)+{\delta }_4{I}_4\left(\mathfrak{G}\right)+\mu P_2\left(\mathfrak{G}\right)-(\alpha +d)S\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill I_1\left(\mathfrak{Z}\right)& =& \breve{I_1}+{\int }_0^{\mathfrak{Z}}\left({\eta }_2+{\xi }_{1}S\left(\mathfrak{G}\right)I_1\left(\mathfrak{G}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill I_2\left(\mathfrak{Z}\right)& =& \breve{I_2}+{\int }_0^{\mathfrak{Z}}\left({\eta }_3+{\xi }_{2}S\left(\mathfrak{G}\right)I_2\left(\mathfrak{G}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill I_3\left(\mathfrak{Z}\right)& =& \breve{I_3}+{\int }_0^{\mathfrak{Z}}\left({\eta }_4+{\xi }_{3}S\left(\mathfrak{G}\right)I_3\left(\mathfrak{G}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill I_4\left(\mathfrak{Z}\right)& =& \breve{I_4}+{\int }_0^{\mathfrak{Z}}\left({\eta }_5+{\xi }_{4}S\left(\mathfrak{G}\right)I_4\left(\mathfrak{G}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill Q_1\left(\mathfrak{Z}\right)& =& \breve{Q_1}+{\int }_0^{\mathfrak{Z}}\left({\eta }_6+g_1{I}_1\left(\mathfrak{G}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill Q_2\left(\mathfrak{Z}\right)& =& \breve{Q_2}+{\int }_0^{\mathfrak{Z}}\left({\eta }_7+g_2{I}_2\left(\mathfrak{G}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill Q_3\left(\mathfrak{Z}\right)& =& \breve{Q_3}+{\int }_0^{\mathfrak{Z}}\left({\eta }_8+g_3{I}_3\left(\mathfrak{G}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill Q_4\left(\mathfrak{Z}\right)& =& \breve{I_2}+{\int }_0^{\mathfrak{Z}}\left({\eta }_9+g_4{I}_4\left(\mathfrak{G}\right)-({\Theta }_4+d)Q_4\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill T\left(\mathfrak{Z}\right)& =& \breve{I_2}+{\int }_0^{\mathfrak{Z}}({\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{G}\right)+{\Theta }_2{Q}_2\left(\mathfrak{G}\right)+{\Theta }_3{Q}_3\left(\mathfrak{G}\right)+{\Theta }_4{Q}_4\left(\mathfrak{G}\right)\hfill \\ & -& (a_1+a_2+d)T\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill P_1\left(\mathfrak{Z}\right)& =& \breve{I_2}+{\int }_0^{\mathfrak{Z}}\left({\eta }_{11}+a_{1}T\left(\mathfrak{G}\right)-(\Delta +d)P_1\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \\ \hfill P_2\left(\mathfrak{Z}\right)& =& \breve{I_2}+{\int }_0^{\mathfrak{Z}}\left({\eta }_{12}+a_{2}T\left(\mathfrak{G}\right)+\Delta P_1\left(\mathfrak{G}\right)+\alpha S\left(\mathfrak{G}\right)-(\mu +d)P_2\left(\mathfrak{G}\right)\right)d\mathfrak{G}\hfill \end{array}$$

(3)

For convenience, the kernels listed below are defined:

$$\begin{array}{ccc}\hfill {\aleph }_1(\mathfrak{Z}, S\left(\mathfrak{Z}\right))& =& {\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_2(\mathfrak{Z}, I_1\left(\mathfrak{Z}\right))& =& {\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_3(\mathfrak{Z}, I_2\left(\mathfrak{Z}\right))& =& {\eta }_3+{\xi }_{2}S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_4(\mathfrak{Z}, I_3\left(\mathfrak{Z}\right))& =& {\eta }_4+{\xi }_{3}S\left(\mathfrak{Z}\right)I_3\left(\mathfrak{Z}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_5(\mathfrak{Z}, I_4\left(\mathfrak{Z}\right))& =& {\eta }_5+{\xi }_{4}S\left(\mathfrak{Z}\right)I_4\left(\mathfrak{Z}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_6(\mathfrak{Z}, Q_1\left(\mathfrak{Z}\right))& =& {\eta }_6+g_1{I}_1\left(\mathfrak{Z}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_7(\mathfrak{Z}, Q_2\left(\mathfrak{Z}\right))& =& {\eta }_7+g_2{I}_2\left(\mathfrak{Z}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_8(\mathfrak{Z}, Q_3\left(\mathfrak{Z}\right))& =& {\eta }_8+g_3{I}_3\left(\mathfrak{Z}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_9(\mathfrak{Z}, Q_4\left(\mathfrak{Z}\right))& =& {\eta }_9+g_4{I}_4\left(\mathfrak{Z}\right)-({\Theta }_4+d)Q_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_{10}(\mathfrak{Z}, T\left(\mathfrak{Z}\right))& =& {\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{Z}\right)+{\Theta }_2{Q}_2\left(\mathfrak{Z}\right)+{\Theta }_3{Q}_3\left(\mathfrak{Z}\right)+{\Theta }_4{Q}_4\left(\mathfrak{Z}\right)-(a_1+a_2+d)T\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_{11}(\mathfrak{Z}, P_1\left(\mathfrak{Z}\right))& =& {\eta }_{11}+a_{1}T\left(\mathfrak{Z}\right)-(\Delta +d)P_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\aleph }_{12}(\mathfrak{Z}, P_2\left(\mathfrak{Z}\right))& =& {\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+\Delta P_1\left(\mathfrak{Z}\right)+\alpha S\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right)\hfill \end{array}$$

Since $S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2$ are non-negative bounded functions, ∃ positive values ${\wp }_k, k={\mathbb{N}}_1^{12}$ such that (s.t.)

$$\begin{array}{ccc}\hfill ∥S∥& \le & {\wp }_1, ∥I_1∥\le {\wp }_2, ∥I_2∥\le {\wp }_3, ∥I_3∥\le {\wp }_4, ∥I_4∥\le {\wp }_5∥Q_1∥\le {\wp }_6, ∥Q_2∥\le {\wp }_7\hfill \\ \hfill ∥Q_3∥& \le & {\wp }_8, ∥Q_4∥\le {\wp }_9, ∥T∥\le {\wp }_{10}, ∥P_1∥\le {\wp }_{11}, ∥P_2∥\le {\wp }_{12}.\hfill \end{array}$$

In light of this, Equation (3) can be expressed as

$$\begin{array}{ccc}\hfill S\left(\mathfrak{Z}\right)& =& \breve{S}+{\int }_0^{\mathfrak{Z}}{\aleph }_1(\mathfrak{G}, S\left(\mathfrak{G}\right))d\mathfrak{G}, I_1\left(\mathfrak{Z}\right)=\breve{I_1}+{\int }_0^{\mathfrak{Z}}{\aleph }_2(\mathfrak{G}, I_1\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill I_2\left(\mathfrak{Z}\right)& =& \breve{I_2}+{\int }_0^{\mathfrak{Z}}{\aleph }_3(\mathfrak{G}, I_2\left(\mathfrak{G}\right))d\mathfrak{G}, I_3\left(\mathfrak{Z}\right)=\breve{I_3}+{\int }_0^{\mathfrak{Z}}{\aleph }_4(\mathfrak{G}, I_3\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill I_4\left(\mathfrak{Z}\right)& =& \breve{I_4}+{\int }_0^{\mathfrak{Z}}{\aleph }_5(\mathfrak{G}, I_4\left(\mathfrak{G}\right))d\mathfrak{G}, Q_1\left(\mathfrak{Z}\right)=\breve{Q_1}+{\int }_0^{\mathfrak{Z}}{\aleph }_6(\mathfrak{G}, Q_1\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill Q_2\left(\mathfrak{Z}\right)& =& \breve{Q_2}+{\int }_0^{\mathfrak{Z}}{\aleph }_7(\mathfrak{G}, Q_2\left(\mathfrak{G}\right))d\mathfrak{G}, Q_3\left(\mathfrak{Z}\right)=\breve{Q_3}+{\int }_0^{\mathfrak{Z}}{\aleph }_8(\mathfrak{G}, Q_3\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill Q_4\left(\mathfrak{Z}\right)& =& \breve{Q_4}+{\int }_0^{\mathfrak{Z}}{\aleph }_9(\mathfrak{G}, Q_4\left(\mathfrak{G}\right))d\mathfrak{G}, T\left(\mathfrak{Z}\right)=\breve{T}+{\int }_0^{\mathfrak{Z}}{\aleph }_{10}(\mathfrak{G}, T\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill P_1\left(\mathfrak{Z}\right)& =& \breve{P_1}+{\int }_0^{\mathfrak{Z}}{\aleph }_{11}(\mathfrak{G}, P_1\left(\mathfrak{G}\right))d\mathfrak{G}, P_2\left(\mathfrak{Z}\right)=\breve{P_2}+{\int }_0^{\mathfrak{Z}}{\aleph }_{12}(\mathfrak{G}, P_2\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \end{array}$$

(4)

Theorem 1.

If $0\le \mathrm{{\rm Y}}<1$, then the kernels ${\aleph }_k, k={\mathbb{N}}_1^{12}$ are contraction mappings since they fulfill the Lipschitz condition, where $\mathrm{{\rm Y}}={max}_{1\le k\le 12}{\Re }_k$.

Proof. 

Take the kernel ${\aleph }_1$ into account. If $S\left(\mathfrak{Z}\right)$ and $\overline{S\left(\mathfrak{Z}\right)}$ are two arbitrary functions, then we obtain

$$\begin{array}{ccc}\hfill ∥{\aleph }_1(\mathfrak{Z}, S\left(\mathfrak{Z}\right))-{\aleph }_1(\mathfrak{Z}, \overline{S\left(\mathfrak{Z}\right)})∥& \le & \left({\xi }_1{\wp }_2+{\xi }_2{\wp }_3+{\xi }_3{\wp }_4+{\xi }_4{\wp }_5+(\alpha +d)\right)∥S\left(\mathfrak{Z}\right)-\overline{S\left(\mathfrak{Z}\right)}∥\hfill \\ & =& {\Re }_1∥S\left(\mathfrak{Z}\right)-\overline{S\left(\mathfrak{Z}\right)}∥\hfill \end{array}$$

For kernels ${\aleph }_k, k={\mathbb{N}}_2^{12}$, comparable outcomes are achievable:

$$\begin{array}{ccc}\hfill ∥{\aleph }_2(\mathfrak{Z}, I_1\left(\mathfrak{Z}\right))-{\aleph }_2(\mathfrak{Z}, \overline{I_1\left(\mathfrak{Z}\right)})∥& \le & \left({\xi }_1{\wp }_1+{\delta }_1+g_1+d\right)∥I_1\left(\mathfrak{Z}\right)-\overline{I_1\left(\mathfrak{Z}\right)}∥={\Re }_2∥I_1\left(\mathfrak{Z}\right)-\overline{I_1\left(\mathfrak{Z}\right)}∥\hfill \\ & ⋮& \\ \hfill ∥{\aleph }_{12}(\mathfrak{Z}, P_2\left(\mathfrak{Z}\right))-{\aleph }_{12}(\mathfrak{Z}, \overline{P_2\left(\mathfrak{Z}\right)})∥& \le & \left(\mu +d\right)∥P_2\left(\mathfrak{Z}\right)-\overline{P_2\left(\mathfrak{Z}\right)}∥={\Re }_{12}∥P_2\left(\mathfrak{Z}\right)-\overline{P_2\left(\mathfrak{Z}\right)}∥\hfill \end{array}$$

Consequently, ${\aleph }_k, k={\mathbb{N}}_1^{12}$, satisfying the Lipschitz condition. Since $0\le \mathrm{{\rm Y}}<1$, the kernels are contraction mappings. □

The subsequent recursive formulae are now presented using Equation (4):

$$\begin{array}{ccc}\hfill S^n\left(\mathfrak{Z}\right)& =& {\int }_0^{\mathfrak{Z}}{\aleph }_1(\mathfrak{G}, S^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}, Q_1^n\left(\mathfrak{Z}\right)={\int }_0^{\mathfrak{Z}}{\aleph }_6(\mathfrak{G}, Q_1^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill I_1^n\left(\mathfrak{Z}\right)& =& {\int }_0^{\mathfrak{Z}}{\aleph }_2(\mathfrak{G}, I_1^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}, Q_2^n\left(\mathfrak{Z}\right)={\int }_0^{\mathfrak{Z}}{\aleph }_7(\mathfrak{G}, Q_2^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ & ⋮& \\ \hfill I_4^n\left(\mathfrak{Z}\right)& =& {\int }_0^{\mathfrak{Z}}{\aleph }_5(\mathfrak{G}, I_4^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}, Q_4^n\left(\mathfrak{Z}\right)={\int }_0^{\mathfrak{Z}}{\aleph }_9(\mathfrak{G}, Q_4^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \\ \hfill T^n\left(\mathfrak{Z}\right)& =& {\int }_0^{\mathfrak{Z}}{\aleph }_{10}(\mathfrak{G}, T^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}, P_1^n\left(\mathfrak{Z}\right)={\int }_0^{\mathfrak{Z}}{\aleph }_{11}(\mathfrak{G}, P_1^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}, \hfill \\ \hfill P_2^n\left(\mathfrak{Z}\right)& =& {\int }_0^{\mathfrak{Z}}{\aleph }_{12}(\mathfrak{G}, P_2^{n-1}\left(\mathfrak{G}\right))d\mathfrak{G}\hfill \end{array}$$

Among the recursive formulae, the following are the subtraction of two successive forms:

$$\begin{array}{ccc}\hfill {\Im }_1^n\left(\mathfrak{Z}\right)& =& S^n\left(\mathfrak{Z}\right)-S^{n-1}\left(\mathfrak{Z}\right)={\int }_0^{\mathfrak{Z}}\left({\aleph }_1(\mathfrak{G}, S^{n-1}\left(\mathfrak{G}\right))-{\aleph }_1(\mathfrak{G}, S^{n-2}\left(\mathfrak{G}\right))\right)d\mathfrak{G}\hfill \\ \hfill {\Im }_2^n\left(\mathfrak{Z}\right)& =& I_1^n\left(\mathfrak{Z}\right)-I_1^{n-1}\left(\mathfrak{Z}\right)={\int }_0^{\mathfrak{Z}}\left({\aleph }_2(\mathfrak{G}, I_1^{n-1}\left(\mathfrak{G}\right))-{\aleph }_2(\mathfrak{G}, I_1^{n-2}\left(\mathfrak{G}\right))\right)d\mathfrak{G}\hfill \\ & ⋮& \hfill \\ \hfill {\Im }_{12}^n\left(\mathfrak{Z}\right)& =& P_2^n\left(\mathfrak{Z}\right)-P_2^{n-1}\left(\mathfrak{Z}\right)={\int }_0^{\mathfrak{Z}}\left({\aleph }_{12}(\mathfrak{G}, P_2^{n-1}\left(\mathfrak{G}\right))-{\aleph }_{12}(\mathfrak{G}, P_2^{n-2}\left(\mathfrak{G}\right))\right)d\mathfrak{G}\hfill \end{array}$$

(5)

Consequently, it may be claimed that

$$\begin{array}{ccc}\hfill S^n\left(\mathfrak{Z}\right)& =& \sum _{k=1}^n{\Im }_1^k\left(\mathfrak{Z}\right), I_1^n\left(\mathfrak{Z}\right)=\sum _{k=1}^n{\Im }_2^k\left(\mathfrak{Z}\right),\cdots , P_2^n\left(\mathfrak{Z}\right)=\sum _{k=1}^n{\Im }_{12}^k\left(\mathfrak{Z}\right)\hfill \end{array}$$

(6)

Now,

$$\begin{array}{ccc}\hfill ∥{\Im }_1^n\left(\mathfrak{Z}\right)∥& =& ∥S^n\left(\mathfrak{Z}\right)-S^{n-1}\left(\mathfrak{Z}\right)∥=∥{\int }_0^{\mathfrak{Z}}\left({\aleph }_1(\mathfrak{G}, S^{n-1}\left(\mathfrak{G}\right))-{\aleph }_1(\mathfrak{G}, S^{n-2}\left(\mathfrak{G}\right))\right)d\mathfrak{G}∥\hfill \\ & \le & {\int }_0^{\mathfrak{Z}}∥{\aleph }_1(\mathfrak{G}, S^{n-1}\left(\mathfrak{G}\right))-{\aleph }_1(\mathfrak{G}, S^{n-2}\left(\mathfrak{G}\right))∥d\mathfrak{G}\le {\Re }_1\mathfrak{z}∥S^{n-1}\left(\mathfrak{Z}\right)-S^{n-2}\left(\mathfrak{Z}\right)∥\hfill \\ & & ={\Re }_1\mathfrak{Z}∥{\Im }_1^{n-1}\left(\mathfrak{Z}\right)∥\hfill \end{array}$$

(7)

Similarly,

$$\begin{array}{ccc}\hfill ∥{\Im }_2^n\left(\mathfrak{Z}\right)∥& \le & {\Re }_2\mathfrak{Z}∥{\Im }_2^{n-1}\left(\mathfrak{Z}\right)∥, ∥{\Im }_3^n\left(\mathfrak{Z}\right)∥\le {\Re }_3\mathfrak{Z}∥{\Im }_3^{n-1}\left(\mathfrak{Z}\right)∥, ∥{\Im }_4^n\left(\mathfrak{Z}\right)∥\le {\Re }_4\mathfrak{Z}∥{\Im }_4^{n-1}\left(\mathfrak{Z}\right)∥\hfill \\ \hfill ∥{\Im }_5^n\left(\mathfrak{Z}\right)∥& \le & {\Re }_5\mathfrak{Z}∥{\Im }_5^{n-1}\left(\mathfrak{Z}\right)∥, ∥{\Im }_6^n\left(\mathfrak{Z}\right)∥\le {\Re }_6\mathfrak{Z}∥{\Im }_6^{n-1}\left(\mathfrak{Z}\right)∥, ∥{\Im }_7^n\left(\mathfrak{Z}\right)∥\le {\Re }_7\mathfrak{Z}∥{\Im }_7^{n-1}\left(\mathfrak{Z}\right)∥\hfill \\ \hfill ∥{\Im }_8^n\left(\mathfrak{Z}\right)∥& \le & {\Re }_8\mathfrak{Z}∥{\Im }_8^{n-1}\left(\mathfrak{Z}\right)∥, ∥{\Im }_9^n\left(\mathfrak{Z}\right)∥\le {\Re }_9\mathfrak{Z}∥{\Im }_9^{n-1}\left(\mathfrak{Z}\right)∥, ∥{\Im }_{10}^n\left(\mathfrak{Z}\right)∥\le {\Re }_{10}\mathfrak{Z}∥{\Im }_{10}^{n-1}\left(\mathfrak{Z}\right)∥\hfill \\ \hfill ∥{\Im }_{11}^n\left(\mathfrak{Z}\right)∥& \le & {\Re }_{11}\mathfrak{Z}∥{\Im }_{11}^{n-1}\left(\mathfrak{Z}\right)∥, ∥{\Im }_{12}^n\left(\mathfrak{Z}\right)∥\le {\Re }_{12}\mathfrak{Z}∥{\Im }_{12}^{n-1}\left(\mathfrak{Z}\right)∥\hfill \end{array}$$

(8)

Theorem 2.

A unique solution for model (1) occurs if, for a period ${\mathcal{T}}_0>0$, the following condition holds:

$$0<{\Re }_k{\mathcal{T}}_0<1, k={\mathbb{N}}_1^{12}.$$

Proof. 

Two items comprise the proof:

• Existence. Model (1)’s functions are bounded, and the supplied kernels meet the Lipschitz requirements. Consequently, using (6), the subsequent inequalities may be obtained:

  $$\begin{array}{ccc}\hfill ∥{\Im }_1^n\left(\mathfrak{Z}\right)∥& \le & {\Re }_1\mathfrak{Z}∥{\Im }_1^{n-1}\left(\mathfrak{Z}\right)∥\le {\left({\Re }_1\mathfrak{Z}\right)}^2∥{\Im }_1^{n-2}\left(\mathfrak{Z}\right)∥\le \cdots \le {\left({\Re }_1\mathfrak{Z}\right)}^n∥\breve{S}∥\hfill \\ \hfill ∥{\Im }_2^n\left(\mathfrak{Z}\right)∥& \le & {\Re }_2\mathfrak{Z}∥{\Im }_2^{n-1}\left(\mathfrak{Z}\right)∥\le {\left({\Re }_2\mathfrak{Z}\right)}^2∥{\Im }_2^{n-2}\left(\mathfrak{Z}\right)∥\le \cdots \le {\left({\Re }_2\mathfrak{Z}\right)}^n∥\breve{I_1}∥\hfill \\ & ⋮& \hfill \\ \hfill ∥{\Im }_{12}^n\left(\mathfrak{Z}\right)∥& \le & {\Re }_{12}\mathfrak{Z}∥{\Im }_{12}^{n-1}\left(\mathfrak{Z}\right)∥\le {\left({\Re }_{12}\mathfrak{Z}\right)}^2∥{\Im }_{12}^{n-2}\left(\mathfrak{Z}\right)∥\le \cdots \le {\left({\Re }_{12}\mathfrak{Z}\right)}^n∥\breve{P_2}∥\hfill \end{array}$$

  (9)

• The smoothness and existence of the functions specified in (6) are confirmed by the inequalities (9). Define ${\Psi }_k^n, k={\mathbb{N}}_1^{12}$ as the remaining terms after n iterations, that is,

  $$\begin{array}{ccc}\hfill S\left(\mathfrak{Z}\right)-\breve{S}& =& S^n\left(\mathfrak{Z}\right)+{\Psi }_1^n, I_1\left(\mathfrak{Z}\right)-\breve{I_1}=I_1^n\left(\mathfrak{Z}\right)+{\Psi }_2^n, I_2\left(\mathfrak{Z}\right)-\breve{I_2}=I_2^n\left(\mathfrak{Z}\right)+{\Psi }_3^n\hfill \\ \hfill I_3\left(\mathfrak{Z}\right)-\breve{I_3}& =& I_3^n\left(\mathfrak{Z}\right)+{\Psi }_4^n, I_4\left(\mathfrak{Z}\right)-\breve{I_4}=I_4^n\left(\mathfrak{Z}\right)+{\Psi }_5^n, Q_1\left(\mathfrak{Z}\right)-\breve{Q_1}=Q_1^n\left(\mathfrak{Z}\right)+{\Psi }_6^n\hfill \\ \hfill Q_2\left(\mathfrak{Z}\right)-\breve{Q_2}& =& Q_2^n\left(\mathfrak{Z}\right)+{\Psi }_7^n, Q_3\left(\mathfrak{Z}\right)-\breve{Q_3}=Q_3^n\left(\mathfrak{Z}\right)+{\Psi }_8^n, Q_4\left(\mathfrak{Z}\right)-\breve{Q_4}=Q_4^n\left(\mathfrak{Z}\right)+{\Psi }_9^n\hfill \\ \hfill T\left(\mathfrak{Z}\right)-\breve{T}& =& T^n\left(\mathfrak{Z}\right)+{\Psi }_{10}^n, P_1\left(\mathfrak{Z}\right)-\breve{P_1}=P_1^n\left(\mathfrak{Z}\right)+{\Psi }_{11}^n, P_2\left(\mathfrak{Z}\right)-\breve{P_2}=P_2^n\left(\mathfrak{Z}\right)+{\Psi }_{12}^n\hfill \end{array}$$

• Using the Lipschitz condition for ${\aleph }_1$, we obtain

$$\begin{array}{ccc}\hfill ∥{\Psi }_1^n\left(\mathfrak{Z}\right)∥& =& ∥{\int }_0^{\mathfrak{Z}}\left({\aleph }_1(\mathfrak{G}, S\left(\mathfrak{G}\right))-{\aleph }_1(\mathfrak{G}, S^{n-1}\left(\mathfrak{G}\right))\right)d\mathfrak{G}∥\hfill \\ & \le & {\Re }_1\mathfrak{Z}∥S\left(\mathfrak{Z}\right)-S^{n-1}\left(\mathfrak{Z}\right)∥\le \cdots \le {\left({\Re }_1\mathfrak{Z}\right)}^n{\wp }_1\hfill \end{array}$$

• Setting $\mathfrak{Z}={\mathcal{T}}_0$, we obtain

$$∥{\Psi }_1^n\left(\mathfrak{Z}\right)∥\le {\left({\Re }_1{\mathcal{T}}_0\right)}^n{\wp }_1$$

(10)

• $∥{\Psi }_1^n\left(\mathfrak{Z}\right)∥\to 0$ is obtained by taking the limit of inequality (10) as $n\to \infty$ and applying the constraint $0<{\Re }_1{\mathcal{T}}_0<1$. This means that ${lim}_{n\to \infty }{S}^n\left(\mathfrak{Z}\right)=S\left(\mathfrak{Z}\right)-\breve{S}$. In a similar manner, the following inequalities are obtained:

$$\begin{array}{ccc}\hfill ∥{\Psi }_2^n\left(\mathfrak{Z}\right)∥& \le & {\left({\Re }_2{\mathcal{T}}_0\right)}^n{\wp }_2, ∥{\Psi }_3^n\left(\mathfrak{Z}\right)∥\le {\left({\Re }_3{\mathcal{T}}_0\right)}^n{\wp }_3, ∥{\Psi }_4^n\left(\mathfrak{Z}\right)∥\le {\left({\Re }_4{\mathcal{T}}_0\right)}^n{\wp }_4, \hfill \\ \hfill ∥{\Psi }_5^n\left(\mathfrak{Z}\right)∥& \le & {\left({\Re }_5{\mathcal{T}}_0\right)}^n{\wp }_5\hfill \\ \hfill ∥{\Psi }_6^n\left(\mathfrak{Z}\right)∥& \le & {\left({\Re }_6{\mathcal{T}}_0\right)}^n{\wp }_6, ∥{\Psi }_7^n\left(\mathfrak{Z}\right)∥\le {\left({\Re }_7{\mathcal{T}}_0\right)}^n{\wp }_7, ∥{\Psi }_8^n\left(\mathfrak{Z}\right)∥\le {\left({\Re }_8{\mathcal{T}}_0\right)}^n{\wp }_8, \hfill \\ \hfill ∥{\Psi }_9^n\left(\mathfrak{Z}\right)∥& \le & {\left({\Re }_9{\mathcal{T}}_0\right)}^n{\wp }_9\hfill \\ \hfill ∥{\Psi }_{10}^n\left(\mathfrak{Z}\right)∥& \le & {\left({\Re }_{10}{\mathcal{T}}_0\right)}^n{\wp }_{10}, ∥{\Psi }_{11}^n\left(\mathfrak{Z}\right)∥\le {\left({\Re }_{11}{\mathcal{T}}_0\right)}^n{\wp }_{11}, ∥{\Psi }_{12}^n\left(\mathfrak{Z}\right)∥\le {\left({\Re }_{12}{\mathcal{T}}_0\right)}^n{\wp }_{12}\hfill \end{array}$$

• Hence, $∥{\Psi }_k^n\left(\mathfrak{Z}\right)∥\to 0, k={\mathbb{N}}_1^{12}$, as $n\to \infty$. System (1)’s existence is thus established.

• Uniqueness. Let $\mathfrak{X}$ and $\mathfrak{Y}$ be the solution sets to the model (1) s.t.

$$\begin{array}{ccc}\hfill \mathfrak{X}& =& \left(S\left(\mathfrak{Z}\right), I_1\left(\mathfrak{Z}\right),\cdots , I_4\left(\mathfrak{Z}\right), Q_1\left(\mathfrak{Z}\right),\cdots , Q_4\left(\mathfrak{Z}\right), T\left(\mathfrak{Z}\right), P_1\left(\mathfrak{Z}\right), P_2\left(\mathfrak{Z}\right)\right)\hfill \\ \hfill \mathfrak{Y}& =& \left(\tilde{S}\left(\mathfrak{Z}\right), \widetilde{I_1}\left(\mathfrak{Z}\right),\cdots , \widetilde{I_4}\left(\mathfrak{Z}\right), \widetilde{Q_1}\left(\mathfrak{Z}\right),\cdots , \widetilde{Q_4}\left(\mathfrak{Z}\right), \tilde{T}\left(\mathfrak{Z}\right), \widetilde{P_1}\left(\mathfrak{Z}\right), \widetilde{P_2}\left(\mathfrak{Z}\right)\right)\hfill \end{array}$$

• Then, using $0<{\Re }_1{\mathcal{T}}_0<1$, we obtain

$$∥S\left(\mathfrak{Z}\right)-\tilde{S}\left(\mathfrak{Z}\right)∥\le {\Re }_1\mathfrak{Z}∥S\left(\mathfrak{Z}\right)-\tilde{S}\left(\mathfrak{Z}\right)∥$$

• So, $(1-{\Re }_1\mathfrak{Z})∥S\left(\mathfrak{Z}\right)-\tilde{S}\left(\mathfrak{Z}\right)∥\le 0$. At last, one obtains $∥S\left(\mathfrak{Z}\right)-\tilde{S}\left(\mathfrak{Z}\right)∥=0$, that is, $S\left(\mathfrak{Z}\right)\equiv \tilde{S}\left(\mathfrak{Z}\right)$. In the same way, we can obtain

$$\begin{array}{ccc}\hfill I_1\left(\mathfrak{Z}\right)& \equiv & \widetilde{I_1}\left(\mathfrak{Z}\right), \cdots , P_2\left(\mathfrak{Z}\right)\equiv \widetilde{P_2}\left(\mathfrak{Z}\right)\hfill \end{array}$$

It is demonstrated that model (1)’s solutions are unique.

□

2.2. Equilibrium Points and Stability Analysis

To calculate the equilibrium points of the proposed deterministic model (1), set the right side of the modeling equations to zero.

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathbf{d}S\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right)=0\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)=0\hfill \\ & ⋮& \hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+\Delta P_1\left(\mathfrak{Z}\right)+\alpha S\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right)=0\hfill \end{array}$$

(11)

The malware-free equilibrium point is provided by

$$M{F}^0\left({\displaystyle \frac{{\eta }_1(\mu +d)+{\eta }_{12}\mu }{d^2+\alpha d+\mu d}}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, {\displaystyle \frac{{\eta }_1\alpha +{\eta }_{12}(\alpha +d)}{d^2+\alpha d+\mu d}}\right).$$

The endemic equilibrium points are represented by

$$M{E}^{*}\left(S^{*}, I_1^{*},\cdots , I_4^{*}, Q_1^{*},\cdots , Q_4^{*}, T^{*}, P_1^{*}, P_2^{*}\right),$$

which may be found by concurrently solving the set of Equation (11). Discussions on the stability of the proposed model with regard to equilibrium points are particularly problematic because of its significantly higher dimension. As a result, we shall examine the stability analysis using ${\mathfrak{R}}_0$ as the basic reproduction number of model given by (1), approached through the next-generation matrix technique [21].

$${\mathfrak{R}}_0=\sum _{k=1}^4{\displaystyle \frac{g_k{\xi }_k({\eta }_1(\mu +d)+{\eta }_{12}\mu )}{({\Theta }_k+d)({\delta }_k+g_k+d)(d^2+\alpha d+\mu d)}}$$

Theorem 3.

The malware-free equilibrium $\left(M{F}^0\right)$ of the deterministic model (1) is locally asymptotically stable if ${\mathfrak{R}}_0<1$ and unstable if ${\mathfrak{R}}_0>1$.

2.3. Analyzing Optimal Control

Malware attacks are starting to have an increasing impact on global economies, resulting in significant direct losses such as labor charges, expenses for removing and fixing malware from systems, lost user productivity, income from malfunctioning or decaying systems, and other costs directly associated with a malware attack. To effectively prevent malware attacks and safeguard against malware threats, both individuals and corporations use protective barriers and regularly update anti-malware software. Nevertheless, these precautionary measures come at a substantial security cost. Our objective is to reduce total investment costs as well as direct losses from security investments, which makes it plausible to claim that the quantity of infected devices correlates with the amount of money lost due to malware. Additionally, we assume that user security awareness can result in maintaining a certain control on the isolated, traced, and semi-protected servers. So, let us define the next optimum control (O.C.) problem to maximize the ensuing objective functional (O.F.):

$$\mathcal{J}(\Delta \left(\mathfrak{Z}\right))={\int }_0^{\mathcal{T}}\left[C_{1}T\left(\mathfrak{Z}\right)+C_2\sum _{k=1}^4{Q}_k\left(\mathfrak{Z}\right)+C_3{P}_1\left(\mathfrak{Z}\right)+{\displaystyle \frac{C_4}{2}}{\Delta }^2\left(\mathfrak{Z}\right)\right]d\mathfrak{Z}$$

(12)

subject to

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathbf{d}S\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_3+{\xi }_{2}S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_4+{\xi }_{3}S\left(\mathfrak{Z}\right)I_3\left(\mathfrak{Z}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_5+{\xi }_{4}S\left(\mathfrak{Z}\right)I_4\left(\mathfrak{Z}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_6+g_1{I}_1\left(\mathfrak{Z}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_7+g_2{I}_2\left(\mathfrak{Z}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_8+g_3{I}_3\left(\mathfrak{Z}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_9+g_4{I}_4\left(\mathfrak{Z}\right)-({\Theta }_4+d)Q_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}T\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{Z}\right)+{\Theta }_2{Q}_2\left(\mathfrak{Z}\right)+{\Theta }_3{Q}_3\left(\mathfrak{Z}\right)+{\Theta }_4{Q}_4\left(\mathfrak{Z}\right)-(a_1+a_2+d)T\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{11}+a_{1}T\left(\mathfrak{Z}\right)-(\Delta +d)P_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+\alpha S\left(\mathfrak{Z}\right)+\Delta P_1\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right)\hfill \end{array}$$

(13)

where $C_1$ is the public awareness coefficient of tracing servers per unit time, $C_2$ is the public awareness coefficient of isolated servers per unit time, $C_3$ is the public awareness coefficient of semi-protected servers per unit time, and $C_4$ is the awareness coefficient of semi-protected servers due to heavy infections, and the total maintained security level is symbolized by $\mathfrak{J}(\Delta )$. The controller objective framework is as indicated in Figure 2.

To obtain an optimum solution for the O.C. problems ((12) and (13)), we need to provide the Lagrangian and Hamiltonian. In actuality, the Lagrangian of the optimal problem is supplied by

$$\mathfrak{L}(T, Q_1,\cdots , Q_4, P_1, \Delta )=C_{1}T\left(\mathfrak{Z}\right)+C_2\sum _{k=1}^4{Q}_k\left(\mathfrak{Z}\right)+C_3{P}_1\left(\mathfrak{Z}\right)+{\displaystyle \frac{C_4}{2}}{\Delta }^2\left(\mathfrak{Z}\right)$$

Next, we need to choose a suitable $\Delta \left(\mathfrak{Z}\right)$ s.t. the Lagrangian integral discussed before achieves its minimum. This may be accomplished by defining the Hamiltonian $\mathfrak{H}$ of the control problem as follows:

$$\begin{array}{ccc}& & \mathfrak{H}(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2, \Delta , {\zeta }_1,\cdots , {\zeta }_{12}, \mathfrak{Z})\hfill \\ & & =\mathfrak{L}(T, Q_1,\cdots , Q_4, P_1, \Delta )\hfill \\ & & +[{\zeta }_1{\displaystyle \frac{\mathbf{d}S\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\zeta }_2{\displaystyle \frac{\mathbf{d}{I}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{z}}}+{\zeta }_3{\displaystyle \frac{\mathbf{d}{I}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{z}}}+{\zeta }_4{\displaystyle \frac{\mathbf{d}{I}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{z}}}+{\zeta }_5{\displaystyle \frac{\mathbf{d}{I}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{z}}}+{\zeta }_6{\displaystyle \frac{\mathbf{d}{Q}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}\hfill \\ & & +{\zeta }_7{\displaystyle \frac{\mathbf{d}{Q}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\zeta }_8{\displaystyle \frac{\mathbf{d}{Q}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\zeta }_9{\displaystyle \frac{\mathbf{d}{Q}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\zeta }_{10}{\displaystyle \frac{\mathbf{d}T\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\zeta }_{11}{\displaystyle \frac{\mathbf{d}{P}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\zeta }_{12}{\displaystyle \frac{\mathbf{d}{P}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}]\hfill \end{array}$$

(14)

with ${\zeta }_k, k={\mathbb{N}}_1^{12}$ representing the adjoint variables that need to be appropriately estimated.

Theorem 4.

For the O.C. problem ((12) and (13)), let $\widehat{S}, \widehat{I_1},\cdots , \widehat{P_1}$ and $\widehat{P_2}$ be the optimal state solutions linked to the O.C. variable $\widehat{\Delta }$. Following that, adjoint variables ${\zeta }_k, k={\mathbb{N}}_1^{12}$ exist and they all fulfill

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathbf{d}{\zeta }_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_1\left(\sum _{k=1}^4{\xi }_k\widehat{I_k}+\alpha +d\right)-{\zeta }_2{\xi }_1\widehat{I_1}-{\zeta }_3{\xi }_2\widehat{I_2}-{\zeta }_4{\xi }_3\widehat{I_3}-{\zeta }_5{\xi }_4\widehat{I_4}-{\zeta }_{12}\alpha \hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_1({\xi }_1\widehat{S}-{\delta }_1)-{\zeta }_2({\xi }_1\widehat{S}-{\delta }_1-g_1-d)-{\zeta }_6{g}_1\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_1({\xi }_2\widehat{S}-{\delta }_2)-{\zeta }_3({\xi }_2\widehat{S}-{\delta }_2-g_2-d)-{\zeta }_7{g}_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_1({\xi }_3\widehat{S}-{\delta }_3)-{\zeta }_4({\xi }_3\widehat{S}-{\delta }_3-g_3-d)-{\zeta }_8{g}_3\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_5\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_1({\xi }_4\widehat{S}-{\delta }_4)-{\zeta }_5({\xi }_4\widehat{S}-{\delta }_4-g_4-d)-{\zeta }_9{g}_4\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_6\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_6({\Theta }_1+d)-{\zeta }_{10}{\Theta }_1-C_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_7\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_7({\Theta }_2+d)-{\zeta }_{10}{\Theta }_2-C_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_8\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_8({\Theta }_3+d)-{\zeta }_{10}{\Theta }_3-C_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_9\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_9({\Theta }_3+d)-{\zeta }_{10}{\Theta }_3-C_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_{10}\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_{10}(a_1+a_2+d)-{\zeta }_{11}{a}_1-{\zeta }_{12}{a}_2-C_1\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_{11}\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_{11}(\Delta +d)-{\zeta }_{12}\Delta -C_3\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_{12}\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\zeta }_{12}(\mu +d)-{\zeta }_1\mu \hfill \end{array}$$

(15)

with

$${\zeta }_k\left(\mathcal{T}\right)=0, k={\mathbb{N}}_1^{12}$$

(16)

Additionally, the optimal control $\widehat{\Delta }\left(\mathfrak{Z}\right)$ is provided by

$$\widehat{\Delta }\left(\mathfrak{Z}\right)=max\left[min\left(0, {\displaystyle \frac{{\zeta }_{11}-{\zeta }_{12}}{C_4}}\widehat{P_1}\right), 1\right]$$

(17)

Proof. 

First, the Hamiltonian (14) is utilized to create the adjoint equations. Let $S\left(\mathfrak{Z}\right)=\widehat{S}\left(\mathfrak{Z}\right)$, $I_1\left(\mathfrak{Z}\right)=\widehat{I_1}\left(\mathfrak{Z}\right)$, ⋯, $P_2\left(\mathfrak{Z}\right)=\widehat{P_2}\left(\mathfrak{Z}\right)$, and we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathbf{d}{\zeta }_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial S}}={\zeta }_1\left(\sum _{k=1}^4{\xi }_k\widehat{I_k}+\alpha +d\right)-{\zeta }_2{\xi }_1\widehat{I_1}-{\zeta }_3{\xi }_2\widehat{I_2}-{\zeta }_4{\xi }_3\widehat{I_3}-{\zeta }_5{\xi }_4\widehat{I_4}-{\zeta }_{12}\alpha \hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial I_1}}={\zeta }_1({\xi }_1\widehat{S}-{\delta }_1)-{\zeta }_2({\xi }_1\widehat{S}-{\delta }_1-g_1-d)-{\zeta }_6{g}_1\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial I_2}}={\zeta }_1({\xi }_2\widehat{S}-{\delta }_2)-{\zeta }_3({\xi }_2\widehat{S}-{\delta }_2-g_2-d)-{\zeta }_7{g}_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial I_3}}={\zeta }_1({\xi }_3\widehat{S}-{\delta }_3)-{\zeta }_4({\xi }_3\widehat{S}-{\delta }_3-g_3-d)-{\zeta }_8{g}_3\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_5\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial I_4}}={\zeta }_1({\xi }_4\widehat{S}-{\delta }_4)-{\zeta }_5({\xi }_4\widehat{S}-{\delta }_4-g_4-d)-{\zeta }_9{g}_4\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_6\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial Q_1}}={\zeta }_6({\Theta }_1+d)-{\zeta }_{10}{\Theta }_1-C_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_7\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial Q_2}}={\zeta }_7({\Theta }_2+d)-{\zeta }_{10}{\Theta }_2-C_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_8\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial Q_3}}={\zeta }_8({\Theta }_3+d)-{\zeta }_{10}{\Theta }_3-C_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_9\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial Q_4}}={\zeta }_9({\Theta }_4+d)-{\zeta }_{10}{\Theta }_4-C_2\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_{10}\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial T}}={\zeta }_{10}(a_1+a_2+d)-{\zeta }_{11}{a}_1-{\zeta }_{12}{a}_2-C_1\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_{11}\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial P_1}}={\zeta }_{11}(\Delta +d)-{\zeta }_{12}\Delta -C_3\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{\zeta }_{12}\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& -{\displaystyle \frac{\partial \mathfrak{H}}{\partial P_2}}={\zeta }_{12}(\mu +d)-{\zeta }_1\mu \hfill \end{array}$$

By virtue of the optimality criteria, we have

$${\displaystyle \frac{\partial \mathfrak{H}}{\partial \Delta }}{\mid }_{\Delta =\widehat{\Delta }}=C_4\widehat{\Delta }\left(\mathfrak{Z}\right)-{\zeta }_{11}\widehat{P_1}\left(\mathfrak{Z}\right)+{\zeta }_{12}\widehat{P_1}\left(\mathfrak{Z}\right)=0$$

The identification indicated above suggests that

$$\widehat{\Delta }\left(\mathfrak{Z}\right)={\displaystyle \frac{{\zeta }_{11}-{\zeta }_{12}}{C_4}}\widehat{P_1}$$

Hence,

$$\widehat{\Delta }\left(\mathfrak{Z}\right)=\left\{\begin{array}{cc}0\hfill & , {\displaystyle \frac{\partial \mathfrak{H}}{\partial \Delta }}>0\hfill \\ {\displaystyle \frac{{\zeta }_{11}-{\zeta }_{12}}{C_4}}\widehat{P_1}\hfill & , {\displaystyle \frac{\partial \mathfrak{H}}{\partial \Delta }}=0\hfill \\ 1\hfill & , {\displaystyle \frac{\partial \mathfrak{H}}{\partial \Delta }}<0\hfill \end{array}\right.$$

Consequently, $\widehat{\Delta }\left(\mathfrak{Z}\right)$ is the optimal control, and it may be represented in the concise notation that follows:

$$\widehat{\Delta }\left(\mathfrak{Z}\right)=max\left[min\left(0, {\displaystyle \frac{{\zeta }_{11}-{\zeta }_{12}}{C_4}}\widehat{P_1}\right), 1\right]$$

Here, the optimum control is characterized by Formula (17) for $\widehat{\Delta }$. The characterization of the optimal control (17), the adjoint system (15) and (16), the state system (13) with boundary conditions, and the optimality system can be solved to yield the optimum control and states. To solve the optimality system, we incorporate the characterization of the optimum control $\widehat{\Delta }$ given by (17) along with the transversality and initial conditions. By changing the $\widehat{\Delta }$ values in the control system, the following system is produced for the state system (13).

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathbf{d}S\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_3+{\xi }_{2}S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_4+{\xi }_{3}S\left(\mathfrak{Z}\right)I_3\left(\mathfrak{Z}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_5+{\xi }_{4}S\left(\mathfrak{Z}\right)I_4\left(\mathfrak{Z}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_6+g_1{I}_1\left(\mathfrak{Z}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_7+g_2{I}_2\left(\mathfrak{Z}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_8+g_3{I}_3\left(\mathfrak{Z}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_9+g_4{I}_4\left(\mathfrak{Z}\right)-({\Theta }_4+d)Q_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}T\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{Z}\right)+{\Theta }_2{Q}_2\left(\mathfrak{Z}\right)+{\Theta }_3{Q}_3\left(\mathfrak{Z}\right)+{\Theta }_4{Q}_4\left(\mathfrak{Z}\right)-(a_1+a_2+d)T\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{11}+a_{1}T\left(\mathfrak{Z}\right)-\left(max\left[min\left(0, {\displaystyle \frac{{\zeta }_{11}-{\zeta }_{12}}{C_4}}\widehat{P_1}\right), 1\right]+d\right)P_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+max\left[min\left(0, {\displaystyle \frac{{\zeta }_{11}-{\zeta }_{12}}{C_4}}\widehat{P_1}\right), 1\right]P_1\left(\mathfrak{Z}\right)\hfill \\ & +& \alpha S\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right)\hfill \end{array}$$

(18)

The goal is to determine the optimal control and state system by numerically solving the aforementioned system (18). □

3. Stochastic MMCE Model

Ecologists and biologists argue that a variety of environmental disturbances, including temperature, water availability, and climate change, inevitably impact the spread of infectious diseases, human contact, human movement, and other processes [22,23,24]. In contrast, the deterministic model mentioned above does not account for the contributions of random variables. Furthermore, as far as continuous-time system modeling is concerned, Brownian motion, or B.M., is the preferred method for modeling random motion and noise. Based on B.M.’s strong statistical features, this decision is well founded. One may solve the B.M. issue with the aid of strong analytical methods, as well as finite moments of all orders and continuous sample-path trajectories. Even complicated systems with a high degree of unpredictability as a result of applied external noise are discussed using stochastic models. We designed the stochastic model to generate realistic curves characterizing malware in multi-cloud configurations. Further originality is brought to our article by this modeling method. One generates the stochastic model by first adding white noise aspects to each of the deterministic model equations. The terminologies that have been defined now encompass a wider spectrum of potential viral dynamics and more probable situations. Following the addition of the additional stochastic components, the resulting model equations are as described in the model ((19) and (20)).

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathbf{d}S\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right)+\kappa S\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)+\kappa I_1\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_3+{\xi }_{2}S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{Z}\right)+\kappa I_2\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_4+{\xi }_{3}S\left(\mathfrak{Z}\right)I_3\left(\mathfrak{Z}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{Z}\right)+\kappa I_3\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{I}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_5+{\xi }_{4}S\left(\mathfrak{Z}\right)I_4\left(\mathfrak{Z}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{Z}\right)+\kappa I_4\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_6+g_1{I}_1\left(\mathfrak{Z}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{Z}\right)+\kappa Q_1\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_7+g_2{I}_2\left(\mathfrak{Z}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{Z}\right)+\kappa Q_2\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_8+g_3{I}_3\left(\mathfrak{Z}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{Z}\right)+\kappa Q3\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{Q}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_9+g_4{I}_4\left(\mathfrak{Z}\right)-({\Theta }_4+d)Q_4\left(\mathfrak{Z}\right)+\kappa Q_4\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}T\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{Z}\right)+{\Theta }_2{Q}_2\left(\mathfrak{Z}\right)+{\Theta }_3{Q}_3\left(\mathfrak{Z}\right)+{\Theta }_4{Q}_4\left(\mathfrak{Z}\right)-(a_1+a_2+d)T\left(\mathfrak{Z}\right)\hfill \\ & +& \kappa T\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{11}+a_{1}T\left(\mathfrak{Z}\right)-(\Delta +d)P_1\left(\mathfrak{Z}\right)+\kappa P_1\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \\ \hfill {\displaystyle \frac{\mathbf{d}{P}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}& =& {\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+\Delta P_1\left(\mathfrak{Z}\right)+\alpha S\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right)+\kappa P_2\left(\mathfrak{Z}\right){\displaystyle \frac{\mathbf{d}\mathfrak{W}}{\mathbf{d}\mathfrak{Z}}}\hfill \end{array}$$

(19)

with initial conditions

$$\begin{array}{ccc}& & S\left(0\right)=S_0, I_1\left(0\right)=I_{1_0},\cdots , P_2\left(0\right)=P_{2_0}\hfill \end{array}$$

(20)

where $\mathfrak{W}\left(\mathfrak{Z}\right)$ is a standard Brownian motion process and $\kappa >0$.

3.1. Existence and Uniqueness of Positive Solution

Theorem 5.

For any initial condition $(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\left(0\right)\in {\mathbb{R}}_{12}^{+}$, there is a unique global positive solution $(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\left(\mathfrak{Z}\right)$ for stochastic model (19), which will be maintained in ${\mathbb{R}}_{12}^{+}$ with probability one.

Proof. 

Since the state variables’ initial conditions are known to satisfy

• $(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\left(0\right)\in {\mathbb{R}}_{12}^{+}$, the coefficients given in Equation (19) are defined and locally Lipschitzian. So, there lies a local unique solution
• $(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\left(\mathfrak{Z}\right)$ of the problem over $\mathfrak{Z}\in [0, {\vartheta }_e)$. In order to determine the solution’s global nature, we must demonstrate that ${\vartheta }_e=\infty$ almost surely (a.s.). Assume that we have a sufficiently large, non-negative number ${\varrho }_0$, and that all initial approximations of the state variables are given within $[{\displaystyle \frac{1}{{\varrho }_0}}, {\varrho }_0]$. For every non-negative integer $\varrho \ge {\varrho }_0$, let the time required to reach completion be given as

$$\begin{array}{ccc}\hfill {\vartheta }_{\varrho }& =& \{\mathfrak{Z}\in [0, {\vartheta }_e):min\left\{(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\left(\mathfrak{Z}\right)\right\}\le {\displaystyle \frac{1}{\varrho }}\hfill \\ & & ormax\left\{(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\left(\mathfrak{Z}\right)\right\}\}\hfill \end{array}$$

the choice of $inf\chi =\infty$ is considered, where $\chi$ is for an empty set. From the result of ${\vartheta }_{\varrho }$, we know that this grows as $\varrho \to \infty$. Set ${\vartheta }_{\infty }={lim}_{\varrho \to \infty }{\vartheta }_{\varrho }$ and ${\vartheta }_e\ge {\vartheta }_{\infty }$ a.s., showing ${\vartheta }_{\infty }=\infty$ a.s. Consider the subsequent non-negative ${\mathcal{C}}^2$-function $\Psi$:

$$\begin{array}{ccc}& & \Psi (S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)=S-1-lnS+I_1-1-ln{I}_1+I_2-1-ln{I}_2\hfill \\ & & +I_3-1-ln{I}_3+I_4-1-ln{I}_4+Q_1-1-ln{Q}_1+Q_2-1-ln{Q}_2+Q_3-1-ln{Q}_3\hfill \\ & & +Q_4-1-ln{Q}_4+T-1-lnT+P_1-1-ln{P}_1+P_2-1-ln{P}_2\hfill \end{array}$$

Utilizing It$\widehat{o}$’s formula yields

$$\begin{array}{ccc}\hfill \mathbf{d}\Psi & =& \mathbb{L}\Psi \mathbf{d}\mathfrak{Z}+\kappa (S-1)\mathbf{d}\mathfrak{W}+\kappa (I_1-1)\mathbf{d}\mathfrak{W}+\kappa (I_2-1)\mathbf{d}\mathfrak{W}+\kappa (I_3-1)\mathbf{d}\mathfrak{W}\hfill \\ & & +\kappa (I_4-1)\mathbf{d}\mathfrak{W}+\kappa (Q_1-1)\mathbf{d}\mathfrak{W}+\kappa (Q_2-1)\mathbf{d}\mathfrak{W}+\kappa (Q_3-1)\mathbf{d}\mathfrak{W}+\kappa (Q_4-1)\mathbf{d}\mathfrak{W}\hfill \\ & & +\kappa (T-1)\mathbf{d}\mathfrak{W}+\kappa (P_1-1)\mathbf{d}\mathfrak{W}+\kappa (P_2-1)\mathbf{d}\mathfrak{W}\hfill \end{array}$$

where

$$\begin{array}{ccc}& & \mathbb{L}\Psi =\left(1-{\displaystyle \frac{1}{S}}\right)({\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right))+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{I_1}}\right)\left({\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{I_2}}\right)\left({\eta }_3+{\xi }_{2}S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{I_3}}\right)\left({\eta }_4+{\xi }_{3}S\left(\mathfrak{Z}\right)I_3\left(\mathfrak{Z}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{I_4}}\right)\left({\eta }_5+{\xi }_{4}S\left(\mathfrak{Z}\right)I_4\left(\mathfrak{Z}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{Q_1}}\right)\left({\eta }_6+g_1{I}_1\left(\mathfrak{Z}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{Q_2}}\right)\left({\eta }_7+g_2{I}_2\left(\mathfrak{Z}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{Q_3}}\right)\left({\eta }_8+g_3{I}_3\left(\mathfrak{Z}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}+\left(1-{\displaystyle \frac{1}{Q_4}}\right)({\eta }_9+g_4{I}_4\left(\mathfrak{Z}\right)\hfill \\ & & -({\Theta }_4+d)Q_4\left(\mathfrak{Z}\right))+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{T}}\right)\left({\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{Z}\right)+{\Theta }_2{Q}_2\left(\mathfrak{Z}\right)+{\Theta }_3{Q}_3\left(\mathfrak{Z}\right)+{\Theta }_4{Q}_4\left(\mathfrak{Z}\right)-(a_1+a_2+d)T\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{P_1}}\right)\left({\eta }_{11}+a_{1}T\left(\mathfrak{Z}\right)-(\Delta +d)P_1\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\hfill \\ & & +\left(1-{\displaystyle \frac{1}{P_2}}\right)\left({\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+\Delta P_1\left(\mathfrak{Z}\right)+\alpha S\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right)\right)+{\displaystyle \frac{{\kappa }^2}{2}}\le \mathfrak{K}, \hfill \end{array}$$

and

$$\mathfrak{K}:=\sum _{k=1}^{12}{\eta }_k+\sum _{k=1}^4\left({\delta }_k+g_k+{\Theta }_k+{\xi }_k\right)+\alpha +\mu +\Delta +a_1+a_2+12d+6{\kappa }^2$$

where the constant $\mathfrak{K}>0$. The approach of proof is the same as in [24], and the remaining proof is omitted. □

3.2. Extinction

Theorem 6.

The solution of model (19) with initial condition $(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\left(0\right)$ can be represented as $(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2)\left(\mathfrak{Z}\right)$; then

$$\begin{array}{ccc}& & \underset{\mathfrak{Z}\to \infty }{lim\; sup}\left(S\left(\mathfrak{Z}\right)+\sum _{k=1}^4{I}_k\left(\mathfrak{Z}\right)+\sum _{k=1}^4{Q}_k\left(\mathfrak{Z}\right)+T\left(\mathfrak{Z}\right)+P_1\left(\mathfrak{Z}\right)+P_2\left(\mathfrak{Z}\right)\right)<\infty , a.s.\hfill \\ & & \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{S\left(\mathfrak{Z}\right)}{\mathfrak{Z}}}=0, \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{I_1\left(\mathfrak{Z}\right)}{\mathfrak{Z}}}=0,\cdots , \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{I_4\left(\mathfrak{Z}\right)}{\mathfrak{Z}}}=0, \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{Q_1\left(\mathfrak{Z}\right)}{\mathfrak{Z}}}=0,\cdots , \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{Q_4\left(\mathfrak{Z}\right)}{\mathfrak{Z}}}=0, \hfill \\ & & \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{T\left(\mathfrak{Z}\right)}{\mathfrak{Z}}}=0, \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{P_1\left(\mathfrak{Z}\right)}{\mathfrak{Z}}}=0, \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{P_2\left(\mathfrak{Z}\right)}{\mathfrak{Z}}}=0, \hfill \end{array}$$

Furthermore, when $d>{\kappa }^2/2$ holds, then

$$\begin{array}{ccc}& & \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{1}{\mathfrak{Z}}}{\int }_0^{\mathfrak{Z}}S\left(\mathfrak{G}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)=0, \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{1}{\mathfrak{Z}}}{\int }_0^{\mathfrak{Z}}{I}_1\left(\mathfrak{G}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)=0,\cdots , \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{1}{\mathfrak{Z}}}{\int }_0^{\mathfrak{Z}}{I}_4\left(\mathfrak{G}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)=0, \hfill \\ & & \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{1}{\mathfrak{Z}}}{\int }_0^{\mathfrak{Z}}{Q}_1\left(\mathfrak{G}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)=0,\cdots , \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{1}{\mathfrak{Z}}}{\int }_0^{\mathfrak{Z}}{Q}_4\left(\mathfrak{G}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)=0, \hfill \\ & & \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{1}{\mathfrak{Z}}}{\int }_0^{\mathfrak{Z}}T\left(\mathfrak{G}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)=0, \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{1}{\mathfrak{Z}}}{\int }_0^{\mathfrak{Z}}{P}_1\left(\mathfrak{G}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)=0, \underset{\mathfrak{Z}\to \infty }{lim}{\displaystyle \frac{1}{\mathfrak{Z}}}{\int }_0^{\mathfrak{Z}}{P}_2\left(\mathfrak{G}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)=0, a.s.\hfill \end{array}$$

Proof. 

The derivation of Theorem 6 is similar to [23]; thus, we skip it. □

3.3. Optimal Control Analysis

In order to avoid complexity, we take a vector of the type

$$\mathbf{X}:={[S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2]}^{Tr}$$

where $Tr$ is the transpose, and

$$\mathbf{d}\mathbf{X}\left(\mathfrak{Z}\right)=\ell (\mathbf{X}, \Delta )\mathbf{d}\mathfrak{Z}+\mathbb{M}\left(\mathbf{X}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)$$

where

$$\begin{array}{ccc}\hfill {\ell }_1(\mathbf{X}, \Delta )& =& {\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_2(\mathbf{X}, \Delta )& =& {\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_3(\mathbf{X}, \Delta )& =& {\eta }_3+{\xi }_{2}S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_4(\mathbf{X}, \Delta )& =& {\eta }_4+{\xi }_{3}S\left(\mathfrak{Z}\right)I_3\left(\mathfrak{Z}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_5(\mathbf{X}, \Delta )& =& {\eta }_5+{\xi }_{4}S\left(\mathfrak{Z}\right)I_4\left(\mathfrak{Z}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_6(\mathbf{X}, \Delta )& =& {\eta }_6+g_1{I}_1\left(\mathfrak{Z}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_7(\mathbf{X}, \Delta )& =& {\eta }_7+g_2{I}_2\left(\mathfrak{Z}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_8(\mathbf{X}, \Delta )& =& {\eta }_8+g_3{I}_3\left(\mathfrak{Z}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_9(\mathbf{X}, \Delta )& =& {\eta }_9+g_4{I}_4\left(\mathfrak{Z}\right)-({\Theta }_4+d)Q_4\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_{10}(\mathbf{X}, \Delta )& =& {\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{Z}\right)+{\Theta }_2{Q}_2\left(\mathfrak{Z}\right)+{\Theta }_3{Q}_3\left(\mathfrak{Z}\right)+{\Theta }_4{Q}_4\left(\mathfrak{Z}\right)-(a_1+a_2+d)T\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_{11}(\mathbf{X}, \Delta )& =& {\eta }_{11}+a_{1}T\left(\mathfrak{Z}\right)-(\Delta +d)P_1\left(\mathfrak{Z}\right)\hfill \\ \hfill {\ell }_{12}(\mathbf{X}, \Delta )& =& {\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+\Delta P_1\left(\mathfrak{Z}\right)+\alpha S\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right), \hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathbb{M}}_1& =& \kappa S, {\mathbb{M}}_2=\kappa I_1, {\mathbb{M}}_3=\kappa I_2, {\mathbb{M}}_4=\kappa I_3, {\mathbb{M}}_5=\kappa I_4, {\mathbb{M}}_6=\kappa Q_1, \hfill \\ \hfill {\mathbb{M}}_7& =& \kappa Q_2, {\mathbb{M}}_8=\kappa Q_3, {\mathbb{M}}_9=\kappa Q_4, {\mathbb{M}}_{10}=\kappa T, {\mathbb{M}}_{11}=\kappa P_1, {\mathbb{M}}_{12}=\kappa P_2\hfill \end{array}$$

We provide an optimum control (O.C.) problem to maximize the ensuing objective functional (O.F.)

$$\mathbb{J}(\Delta \left(\mathfrak{Z}\right))=\mathbb{E}\left\{{\int }_0^{\mathcal{T}}\left[C_{1}T\left(\mathfrak{Z}\right)+C_2\sum _{k=1}^4{Q}_k\left(\mathfrak{Z}\right)+C_3{P}_1\left(\mathfrak{Z}\right)+{\displaystyle \frac{C_4}{2}}{\Delta }^2\left(\mathfrak{Z}\right)\right]d\mathfrak{Z}\right\}$$

(21)

subject to

$$\mathbf{d}\mathbf{X}\left(\mathfrak{Z}\right)=\ell (\mathbf{X}, \Delta )\mathbf{d}\mathfrak{Z}+\mathbb{M}\left(\mathbf{X}\right)\mathbf{d}\mathfrak{W}\left(\mathfrak{Z}\right)$$

(22)

where $C_1$ is the public awareness coefficient of tracing servers per unit time, $C_2$ is the public awareness coefficient of isolated servers per unit time, $C_3$ is the public awareness coefficient of semi-protected servers per unit time, and $C_4$ is the awareness coefficient of semi-protected servers due to heavy infections, and the total maintained security level is symbolized by $\mathbb{J}(\Delta )$. To obtain an optimum solution for the O.C. problems ((21)–(22)), we need to provide the Lagrangian and Hamiltonian. In actuality, the Lagrangian of the optimal problem is supplied by

$$\mathbb{L}(T, Q_1,\cdots , Q_4, P_1, \Delta )=C_{1}T\left(\mathfrak{Z}\right)+C_2\sum _{k=1}^4{Q}_k\left(\mathfrak{Z}\right)+C_3{P}_1\left(\mathfrak{Z}\right)+{\displaystyle \frac{C_4}{2}}{\Delta }^2\left(\mathfrak{Z}\right)$$

Next, we need to choose a suitable $\Delta \left(\mathfrak{Z}\right)$ s.t. the Lagrangian integral discussed before achieves its maximum. This may be accomplished by defining the Hamiltonian $\mathbb{H}$ of the control problem as follows:

$$\begin{array}{ccc}& & \mathbb{H}(S, I_1,\cdots , I_4, Q_1,\cdots , Q_4, T, P_1, P_2, \Delta , {\beta }_1,\cdots \cdots , {\beta }_{12}, {\gamma }_1,\cdots \cdots , {\gamma }_{12}, \mathfrak{Z})\hfill \\ & & =\mathbb{L}(T, Q_1,\cdots , Q_4, P_1, \Delta )\hfill \\ & & +[{\beta }_1{\displaystyle \frac{\mathbf{d}S\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\beta }_2{\displaystyle \frac{\mathbf{d}{I}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{z}}}+{\beta }_3{\displaystyle \frac{\mathbf{d}{I}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{z}}}+{\beta }_4{\displaystyle \frac{\mathbf{d}{I}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{z}}}+{\beta }_5{\displaystyle \frac{\mathbf{d}{I}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{z}}}+{\beta }_6{\displaystyle \frac{\mathbf{d}{Q}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}\hfill \\ & & +{\beta }_7{\displaystyle \frac{\mathbf{d}{Q}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\beta }_8{\displaystyle \frac{\mathbf{d}{Q}_3\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\beta }_9{\displaystyle \frac{\mathbf{d}{Q}_4\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\beta }_{10}{\displaystyle \frac{\mathbf{d}T\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\beta }_{11}{\displaystyle \frac{\mathbf{d}{P}_1\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}+{\beta }_{12}{\displaystyle \frac{\mathbf{d}{P}_2\left(\mathfrak{Z}\right)}{\mathbf{d}\mathfrak{Z}}}]\hfill \\ & & +[\kappa S{\gamma }_1+\kappa I_1{\gamma }_2+\kappa I_2{\gamma }_3+\kappa I_3{\gamma }_4+\kappa I_4{\gamma }_5+\kappa Q_1{\gamma }_6+\kappa Q_2{\gamma }_7+\kappa Q_3{\gamma }_8+\kappa Q_4{\gamma }_9\hfill \\ & & +\kappa T{\gamma }_{10}+\kappa P_1{\gamma }_{11}+\kappa P_2{\gamma }_{12}]\hfill \end{array}$$

(23)

The adjoint variables are represented by ${\beta }_k, k={\mathbb{N}}_1^{12}$ and ${\gamma }_k, k={\mathbb{N}}_1^{12}$.

Theorem 7.

For the O.C. problem ((21)–(22)), let $\widehat{S}, \widehat{I_1},\cdots , \widehat{I_4}, \widehat{Q_1},\cdots , \widehat{Q_4}, \widehat{T}, \widehat{P_1}$, and $\widehat{P_2}$ be the optimal state solutions linked to the O.C. variable $\widehat{\Delta }$. Following that, adjoint variables ${\beta }_k, k={\mathbb{N}}_1^{12}$ exist and they all fulfill

$$\begin{array}{ccc}\hfill \mathbf{d}{\beta }_1\left(\mathfrak{Z}\right)& =& \left[{\beta }_1\left(\sum _{k=1}^4{\xi }_k\widehat{I_k}+\alpha +d\right)-{\beta }_2{\xi }_1\widehat{I_1}-{\beta }_3{\xi }_2\widehat{I_2}-{\beta }_4{\xi }_3\widehat{I_3}-{\beta }_5{\xi }_4\widehat{I_4}-{\beta }_{12}\alpha +\kappa {\gamma }_1\right]\mathbf{d}\mathfrak{Z}\hfill \\ & & +{\gamma }_1\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_2\left(\mathfrak{Z}\right)& =& \left[{\beta }_1({\xi }_1\widehat{S}-{\delta }_1)-{\beta }_2({\xi }_1\widehat{S}-{\delta }_1-g_1-d)-{\beta }_6{g}_1+\kappa {\gamma }_2\right]\mathbf{d}\mathfrak{Z}+{\gamma }_2\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_3\left(\mathfrak{Z}\right)& =& \left[{\beta }_1({\xi }_2\widehat{S}-{\delta }_2)-{\beta }_3({\xi }_2\widehat{S}-{\delta }_2-g_2-d)-{\beta }_7{g}_2+\kappa {\gamma }_3\right]\mathbf{d}\mathfrak{Z}+{\gamma }_3\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_4\left(\mathfrak{Z}\right)& =& \left[{\beta }_1({\xi }_3\widehat{S}-{\delta }_3)-{\beta }_4({\xi }_3\widehat{S}-{\delta }_3-g_3-d)-{\beta }_8{g}_3+\kappa {\gamma }_4\right]\mathbf{d}\mathfrak{Z}+{\gamma }_4\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_5\left(\mathfrak{Z}\right)& =& \left[{\beta }_1({\xi }_4\widehat{S}-{\delta }_4)-{\beta }_5({\xi }_4\widehat{S}-{\delta }_4-g_4-d)-{\beta }_9{g}_4+\kappa {\gamma }_5\right]\mathbf{d}\mathfrak{Z}+{\gamma }_5\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_6\left(\mathfrak{Z}\right)& =& \left[{\beta }_6({\Theta }_1+d)-{\beta }_{10}{\Theta }_1-C_2+\kappa {\gamma }_6\right]\mathbf{d}\mathfrak{Z}+{\gamma }_6\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_7\left(\mathfrak{Z}\right)& =& \left[{\beta }_7({\Theta }_2+d)-{\beta }_{10}{\Theta }_2-C_2+\kappa {\gamma }_7\right]\mathbf{d}\mathfrak{Z}+{\gamma }_7\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_8\left(\mathfrak{Z}\right)& =& \left[{\beta }_8({\Theta }_3+d)-{\beta }_{10}{\Theta }_3-C_2+\kappa {\gamma }_8\right]\mathbf{d}\mathfrak{Z}+{\gamma }_8\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_9\left(\mathfrak{Z}\right)& =& \left[{\beta }_9({\Theta }_4+d)-{\beta }_{10}{\Theta }_4-C_2+\kappa {\gamma }_9\right]\mathbf{d}\mathfrak{Z}+{\gamma }_9\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_{10}\left(\mathfrak{Z}\right)& =& \left[{\beta }_{10}(a_1+a_2+d)-{\beta }_{11}{a}_1-{\beta }_{12}{a}_2-C_1+\kappa {\gamma }_{10}\right]\mathbf{d}\mathfrak{Z}+{\gamma }_{10}\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_{11}\left(\mathfrak{Z}\right)& =& \left[{\beta }_{11}(\Delta +d)-{\beta }_{12}\Delta -C_3+\kappa {\gamma }_{11}\right]\mathbf{d}\mathfrak{Z}+{\gamma }_{11}\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_{12}\left(\mathfrak{Z}\right)& =& \left[{\beta }_{12}(\mu +d)-{\beta }_1\mu +\kappa {\gamma }_{12}\right]\mathbf{d}\mathfrak{Z}+{\gamma }_{12}\mathbf{d}\mathfrak{W}\hfill \end{array}$$

(24)

with

$${\beta }_k\left(\mathcal{T}\right)=0, k={\mathbb{N}}_1^{12}$$

(25)

Additionally, the optimal control $\widehat{\Delta }\left(\mathfrak{Z}\right)$ is provided by

$$\widehat{\Delta }\left(\mathfrak{Z}\right)=max\left[min\left(0, {\displaystyle \frac{{\beta }_{11}-{\beta }_{12}}{C_4}}\widehat{P_1}\right), 1\right]$$

(26)

Proof. 

First, the Hamiltonian (23) is utilized to create the adjoint equations. Let $S\left(\mathfrak{Z}\right)=\widehat{S}\left(\mathfrak{Z}\right)$, $I_1\left(\mathfrak{Z}\right)=\widehat{I_1}\left(\mathfrak{Z}\right)$, ⋯, $P_2\left(\mathfrak{Z}\right)=\widehat{P_2}\left(\mathfrak{Z}\right)$, and we obtain

$$\begin{array}{ccc}\hfill \mathbf{d}{\beta }_1\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial S}}\mathbf{d}\mathfrak{Z}+{\gamma }_1\mathbf{d}\mathfrak{W}=[{\beta }_1\left(\sum _{k=1}^4{\xi }_k\widehat{I_k}+\alpha +d\right)-{\beta }_2{\xi }_1\widehat{I_1}-{\beta }_3{\xi }_2\widehat{I_2}-{\beta }_4{\xi }_3\widehat{I_3}-{\beta }_5{\xi }_4\widehat{I_4}\hfill \\ & & -{\beta }_{12}\alpha +\kappa {\gamma }_1]\mathbf{d}\mathfrak{Z}+{\gamma }_1\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_2\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial I_1}}\mathbf{d}\mathfrak{Z}+{\gamma }_2\mathbf{d}\mathfrak{W}=\left[{\beta }_1({\xi }_1\widehat{S}-{\delta }_1)-{\beta }_2({\xi }_1\widehat{S}-{\delta }_1-g_1-d)-{\beta }_6{g}_1+\kappa {\gamma }_2\right]\mathbf{d}\mathfrak{Z}\hfill \\ & & +{\gamma }_2\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_3\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial I_2}}\mathbf{d}\mathfrak{Z}+{\gamma }_3\mathbf{d}\mathfrak{W}=\left[{\beta }_1({\xi }_2\widehat{S}-{\delta }_2)-{\beta }_3({\xi }_2\widehat{S}-{\delta }_2-g_2-d)-{\beta }_7{g}_2+\kappa {\gamma }_3\right]\mathbf{d}\mathfrak{Z}\hfill \\ & & +{\gamma }_3\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_4\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial I_3}}\mathbf{d}\mathfrak{Z}+{\gamma }_4\mathbf{d}\mathfrak{W}=\left[{\beta }_1({\xi }_3\widehat{S}-{\delta }_3)-{\beta }_4({\xi }_3\widehat{S}-{\delta }_3-g_3-d)-{\beta }_8{g}_3+\kappa {\gamma }_4\right]\mathbf{d}\mathfrak{Z}\hfill \\ & & +{\gamma }_4\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_5\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial I_4}}\mathbf{d}\mathfrak{Z}+{\gamma }_5\mathbf{d}\mathfrak{W}=\left[{\beta }_1({\xi }_4\widehat{S}-{\delta }_4)-{\beta }_5({\xi }_4\widehat{S}-{\delta }_4-g_4-d)-{\beta }_9{g}_4+\kappa {\gamma }_5\right]\mathbf{d}\mathfrak{Z}\hfill \\ & & +{\gamma }_5\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_6\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial Q_1}}\mathbf{d}\mathfrak{Z}+{\gamma }_6\mathbf{d}\mathfrak{W}=\left[{\beta }_6({\Theta }_1+d)-{\beta }_{10}{\Theta }_1-C_2+\kappa {\gamma }_6\right]\mathbf{d}\mathfrak{Z}+{\gamma }_6\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_7\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial Q_2}}\mathbf{d}\mathfrak{Z}+{\gamma }_7\mathbf{d}\mathfrak{W}=\left[{\beta }_7({\Theta }_2+d)-{\beta }_{10}{\Theta }_2-C_2+\kappa {\gamma }_7\right]\mathbf{d}\mathfrak{Z}+{\gamma }_7\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_8\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial Q_3}}\mathbf{d}\mathfrak{Z}+{\gamma }_8\mathbf{d}\mathfrak{W}=\left[{\beta }_8({\Theta }_3+d)-{\beta }_{10}{\Theta }_3-C_2+\kappa {\gamma }_8\right]\mathbf{d}\mathfrak{Z}+{\gamma }_8\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_9\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial Q_4}}\mathbf{d}\mathfrak{Z}+{\gamma }_9\mathbf{d}\mathfrak{W}=\left[{\beta }_9({\Theta }_3+d)-{\beta }_{10}{\Theta }_3-C_2+\kappa {\gamma }_9\right]\mathbf{d}\mathfrak{Z}+{\gamma }_9\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_{10}\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial T}}\mathbf{d}\mathfrak{Z}+{\gamma }_{10}\mathbf{d}\mathfrak{W}=\left[{\beta }_{10}(a_1+a_2+d)-{\beta }_{11}{a}_1-{\beta }_{12}{a}_2-C_1+\kappa {\gamma }_{10}\right]\mathbf{d}\mathfrak{Z}+{\gamma }_{10}\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_{11}\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial P_1}}\mathbf{d}\mathfrak{Z}+{\gamma }_{11}\mathbf{d}\mathfrak{W}=\left[{\beta }_{11}(\Delta +d)-{\beta }_{12}\Delta -C_3+\kappa {\gamma }_{11}\right]\mathbf{d}\mathfrak{Z}+{\gamma }_{11}\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{\beta }_{12}\left(\mathfrak{Z}\right)& =& -{\displaystyle \frac{\partial \mathbb{H}}{\partial P_2}}\mathbf{d}\mathfrak{Z}+{\gamma }_{12}\mathbf{d}\mathfrak{W}=\left[{\beta }_{12}(\mu +d)-{\beta }_1\mu +\kappa {\gamma }_{12}\right]\mathbf{d}\mathfrak{Z}+{\gamma }_{12}\mathbf{d}\mathfrak{W}\hfill \end{array}$$

As a consequence of the optimality criteria, we have

$${\displaystyle \frac{\partial \mathbb{H}}{\partial \Delta }}{\mid }_{\Delta =\widehat{\Delta }}=C_4\widehat{\Delta }\left(\mathfrak{Z}\right)-{\beta }_{11}\widehat{P_1}\left(\mathfrak{Z}\right)+{\beta }_{12}\widehat{P_1}\left(\mathfrak{Z}\right)=0$$

The identification indicated above suggests that

$$\widehat{\Delta }\left(\mathfrak{Z}\right)={\displaystyle \frac{{\beta }_{11}-{\beta }_{12}}{C_4}}\widehat{P_1}$$

Hence,

$$\widehat{\Delta }\left(\mathfrak{Z}\right)=\left\{\begin{array}{cc}0\hfill & , {\displaystyle \frac{\partial \mathbb{H}}{\partial \Delta }}>0\hfill \\ {\displaystyle \frac{{\beta }_{11}-{\beta }_{12}}{C_4}}\widehat{P_1}\hfill & , {\displaystyle \frac{\partial \mathbb{H}}{\partial \Delta }}=0\hfill \\ 1\hfill & , {\displaystyle \frac{\partial \mathbb{H}}{\partial \Delta }}<0\hfill \end{array}\right.$$

Consequently, $\widehat{\Delta }\left(\mathfrak{Z}\right)$ is the optimal control, and it may be represented in the concise notation that follows:

$$\widehat{\Delta }\left(\mathfrak{Z}\right)=max\left[min\left(0, {\displaystyle \frac{{\beta }_{11}-{\beta }_{12}}{C_4}}\widehat{P_1}\right), 1\right]$$

Here, the optimum control is characterized by Formula (26) for $\widehat{\Delta }$. The characterization of the optimal control (26), the adjoint system (24) and (25), the state system (22) with boundary conditions, and the optimality system can be solved to yield the optimum control and states. To solve the optimality system, we incorporate the characterization of the optimum control $\widehat{\Delta }$ given by (26) along with the transversality and initial conditions. By changing the $\widehat{\Delta }$ values in the control system, the following system is produced for the state system (22).

$$\begin{array}{ccc}\hfill \mathbf{d}S\left(\mathfrak{Z}\right)& =& [{\eta }_1-{\xi }_1{I}_1\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_2{I}_2\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_3{I}_3\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)-{\xi }_4{I}_4\left(\mathfrak{Z}\right)S\left(\mathfrak{Z}\right)+{\delta }_1{I}_1\left(\mathfrak{Z}\right)\hfill \\ & & +{\delta }_2{I}_2\left(\mathfrak{Z}\right)+{\delta }_3{I}_3\left(\mathfrak{Z}\right)+{\delta }_4{I}_4\left(\mathfrak{Z}\right)+\mu P_2\left(\mathfrak{Z}\right)-(\alpha +d)S\left(\mathfrak{Z}\right)]\mathbf{d}\mathfrak{Z}+\kappa S\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{I}_1\left(\mathfrak{Z}\right)& =& \left[{\eta }_2+{\xi }_{1}S\left(\mathfrak{Z}\right)I_1\left(\mathfrak{Z}\right)-({\delta }_1+g_1+d)I_1\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa I_1\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{I}_2\left(\mathfrak{Z}\right)& =& \left[{\eta }_3+{\xi }_{2}S\left(\mathfrak{Z}\right)I_2\left(\mathfrak{Z}\right)-({\delta }_2+g_2+d)I_2\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa I_2\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{I}_3\left(\mathfrak{Z}\right)& =& \left[{\eta }_4+{\xi }_{3}S\left(\mathfrak{Z}\right)I_3\left(\mathfrak{Z}\right)-({\delta }_3+g_3+d)I_3\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa I_3\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{I}_4\left(\mathfrak{Z}\right)& =& \left[{\eta }_5+{\xi }_{4}S\left(\mathfrak{Z}\right)I_4\left(\mathfrak{Z}\right)-({\delta }_4+g_4+d)I_4\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa I_4\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{Q}_1\left(\mathfrak{Z}\right)& =& \left[{\eta }_6+g_1{I}_1\left(\mathfrak{Z}\right)-({\Theta }_1+d)Q_1\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa Q_1\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{Q}_2\left(\mathfrak{Z}\right)& =& \left[{\eta }_7+g_2{I}_2\left(\mathfrak{Z}\right)-({\Theta }_2+d)Q_2\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa Q_2\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{Q}_3\left(\mathfrak{Z}\right)& =& \left[{\eta }_8+g_3{I}_3\left(\mathfrak{Z}\right)-({\Theta }_3+d)Q_3\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa Q3\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{Q}_4\left(\mathfrak{Z}\right)& =& \left[{\eta }_9+g_4{I}_4\left(\mathfrak{Z}\right)-({\Theta }_4+d)Q_4\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa Q_4\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}T\left(\mathfrak{Z}\right)& =& \left[{\eta }_{10}+{\Theta }_1{Q}_1\left(\mathfrak{Z}\right)+{\Theta }_2{Q}_2\left(\mathfrak{Z}\right)+{\Theta }_3{Q}_3\left(\mathfrak{Z}\right)+{\Theta }_4{Q}_4\left(\mathfrak{Z}\right)-(a_1+a_2+d)T\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}\hfill \\ & & +\kappa T\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{P}_1\left(\mathfrak{Z}\right)& =& \left[{\eta }_{11}+a_{1}T\left(\mathfrak{Z}\right)-\left(max\left[min\left(0, {\displaystyle \frac{{\beta }_{11}-{\beta }_{12}}{C_4}}\widehat{P_1}\right), 1\right]+d\right)P_1\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}+\kappa P_1\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \\ \hfill \mathbf{d}{P}_2\left(\mathfrak{Z}\right)& =& \left[{\eta }_{12}+a_{2}T\left(\mathfrak{Z}\right)+max\left[min\left(0, {\displaystyle \frac{{\beta }_{11}-{\beta }_{12}}{C_4}}\widehat{P_1}\right), 1\right]P_1\left(\mathfrak{Z}\right)+\alpha S\left(\mathfrak{Z}\right)-(\mu +d)P_2\left(\mathfrak{Z}\right)\right]\mathbf{d}\mathfrak{Z}\hfill \\ & & +\kappa P_2\left(\mathfrak{Z}\right)\mathbf{d}\mathfrak{W}\hfill \end{array}$$

(27)

The goal is to determine the optimal control and state system by numerically solving the aforementioned system (27). □

4. Results and Conclusions

4.1. Results Analysis

The deterministic MMCE model and Stochastic MMCE are numerically simulated using Euler and Euler–Maruyama schemes, respectively. The used time step takes the value 0.02, and the initial assumed values of the twelve dynamic servers states take the values 1000, 20, 25, 27, 35, 30, 18, 15, 23, 60, 40, and 80 for both suggested models, while the transmission rates ${\eta }_1$, ${\eta }_2$, ${\eta }_3$, ${\eta }_4$, ${\eta }_5$, ${\eta }_6$, ${\eta }_7$, ${\eta }_8$, ${\eta }_9$, ${\eta }_{10}$, ${\eta }_{11}$, ${\eta }_{12}$, ${\xi }_1$, ${\xi }_2$, ${\xi }_3$, ${\xi }_4$, ${\delta }_1$, ${\delta }_2$, ${\delta }_3$, ${\delta }_4$, $g_1$, $g_2$, $g_3$, $g_4$, ${\Theta }_1$, ${\Theta }_2$, ${\Theta }_3$, ${\Theta }_4$, $a_1$, $a_2$, $\Delta$, $\mu$, $\alpha$, d, and $\kappa$ take the values 15, 5, 4, 3, 6, 7, 8, 10, 9, 3, 5, 8, 0.00013, 0.00012, 0.0001, 0.0001, 0.05, 0.05, 0.05, 0.05, 0.1, 0.15, 0.08, 0.06, 0.0244, 0.03, 0.02, 0.02, 0.1, 0.05, 0.07, 0.0365, 0.015, 0.025, and 0.035. These parameters values are obtained through trial-and-error techniques to visualize the models’ behaviors. MATLAB software version 2021a is used to run all simulations.

The twelve dynamic states of the network that were produced by first simulating the deterministic MMCE model without taking any control into account are shown in Figure 3. It is clear that susceptible servers are presenting the higher core of total servers with a settling value 789, which means that a lot of servers will be at risk of exposure to infection. Following that, the protected servers have a settling value of about 700 servers. The four infected servers’ categories—healthcare, educational, financial, and media servers—had settling values of 69, 30, 40, and 107, respectively, while the isolated servers of the same four categories had settling values of 240, 229, 292, and 341, respectively. The traced, semi-protected, and protected servers of all categories settled at 162, 222, and 706, respectively. The deterministic controlled MMCE model results are as shown in Figure 4. The susceptible server numbers decreased to a settling value 648 with a reduction of 18%. The four infected servers’ categories settling values reduced to 55, 27, 33, and 85, with a total reduction of 20%, while the isolated servers of the same four categories had reduced settling values of 254, 220, 281, 315, with a total reduction of 3%. The traced servers had a value of 159, with a reduction of 2%. The semi-protected servers of the four categories increased significantly, by 263%, reaching a settling value of 807 servers. So, the controller action succeeded in increasing the total number of servers in the traced, isolated, and semi-protected states and decreased the total number of servers in the susceptible and infected states.

The stochastic and stochastic controlled MMCE models results are as shown in Figure 5 and Figure 6, respectively. The stochastic modeling process provides a more realistic representation of the servers’ state dynamics. Starting by analyzing Figure 5, the susceptible and protected servers settled at a value of 978, while the four infected servers’ categories settled at 101, 41, 54, and 153. And after control, as shown in Figure 6, the susceptible reduced to 717 servers with a reduction of 27%. In addition, the infected servers reduced to the settle values of 62, 35, 37, and 97. The semi-protected servers of the four categories increased significantly, by 166%, from a settle value of 325, as shown in Figure 5, to a controlled value of 863, as shown in Figure 6. So, the controller action also succeeded in maintaining a higher level of security in the stochastic MMCE by increasing the total number of servers on the traced, isolated, and semi-protected states and decreasing the total number of servers in the susceptible and infected states. Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show a visualization comparison of the susceptible, isolated, traced, and semi-protected server dynamics of all supposed models under study with and without control.

Both of the controlled dynamic models in this study have the limitation that all transmission rates between the six different types of virtual machines (VMs) and their variety among media, financial, health, and educational servers have fixed values. The four public awareness coefficients were also taken into consideration as constants inside the maximizing objective functional under investigation when the control framework was imposed. Therefore, when applying these models over longer time periods, both transmission rates and awareness coefficients should be time-dependent to make the description more realistic. And for more generalization, if the controller has a multi-objective framework, modern numerical optimization algorithms can be applied.

4.2. Conclusions

To fully describe multi-cloud environments under malware propagation, this work obtained two state-of-the-art dynamical models: the deterministic MMCE model and the stochastic MMCE model. The dynamical network of system servers’ states is categorized into six different states. Two of these states, isolated and infected, are scattered into four types of servers according to their terms of use: healthcare servers, academic and educational servers, financial services servers, and media servers. Then, a control function is recommended for application in both models to maintain a higher level of security in isolated, traced, and semi-protected states. After running both models with initial state values and using the numerical simulations, the obtained results showed the effectiveness of the recommended control strategy in decreasing the level of susceptible and infected servers, as well as increasing the level of security of the three mentioned states to a significant degree. The semi-protected servers increased by 263% for the deterministic MMCE and 166% for the stochastic MMCE model in terms of their settling values. In our future studies, we will work on modifying the proposed models considering fractional-order stochastic frameworks and then applying them to actual mutli-cloud environments for modeling validations.