Epidemic dynamics of complex networks based on information dependence

Abstract

Vaccination has played a significant role in suppressing the spread of epidemics. In this paper, A Susceptible-Vaccinated-Infected-Recovered (SVIR) epidemic model including degree-dependent transmission rates and imperfect vaccination is proposed. Based on game theory, individuals adopt vaccine and disease information for a period of time to decide whether or not to vaccinate, reflecting the information dependence of individual decisions by introducing a distributed delay. The vaccination rate is determined by the level of epidemic propagation, functioning as a time-varying variable rather than a fixed constant. Explicit expressions for the basic reproduction number and epidemic thresholds related to degree are derived through the next-generation matrix approach. Several sufficient conditions for the existence of five equilibria are presented. Additionally, the stability of the disease-free equilibrium and the persistence of the epidemic are demonstrated. Particularly, considering that individual decisions are influenced only by current information, the global attractivity of the unique endemic equilibrium is verified through the monotone iteration technique. Finally, based on a real contact network from a gallery exhibition in Dublin, we investigate the impact of information-dependent vaccination decisions on epidemic transmission and study the effect of various system parameters on the epidemic thresholds.

Appendices

A Proof of Theorem 2

According to system (4), its Jacobian matrix can be obtained as follows:

$$\begin{aligned} {{\textbf {J}}}=\begin{pmatrix} J_{11}& J_{12}& J_{13}& {{\textbf {0}}}& J_{15}\\ J_{21}& J_{22}& J_{23}& {{\textbf {0}}}& J_{25}\\ J_{31}& J_{32}& J_{33}& {{\textbf {0}}}& {{\textbf {0}}}\\ {{\textbf {0}}}& {{\textbf {0}}}& J_{43}& J_{44}& {{\textbf {0}}}\\ {{\textbf {0}}}& {{\textbf {0}}}& J_{53}& {{\textbf {0}}}& J_{55} \end{pmatrix}, \end{aligned}$$

(15)

where

$$\begin{aligned} & J_{13}=\begin{pmatrix} -\frac{nP(n)\beta ^S(n) S_n}{\left\langle k\right\rangle }& \cdots & -\frac{mP(m)\beta ^S(n) S_n}{\left\langle k\right\rangle }\\ \vdots & \vdots & \vdots \\ -\frac{nP(n)\beta ^S(m) S_m}{\left\langle k\right\rangle }& \cdots & -\frac{mP(m)\beta ^S(m) S_m}{\left\langle k\right\rangle } \end{pmatrix}, \\ & J_{12}=\begin{pmatrix} \delta & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & \delta \end{pmatrix}, J_{15}=\begin{pmatrix} -S_n& \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & -S_m \end{pmatrix}, \\ & J_{21}=\begin{pmatrix} x_n& \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & x_m \end{pmatrix}, J_{25}=\begin{pmatrix} S_n& \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & S_m \end{pmatrix}, \end{aligned}$$

$$\begin{aligned} & J_{22}=\begin{pmatrix} -\Big (\beta ^V(n)\varTheta +\delta +\omega \Big )& \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & -\Big (\beta ^V(m)\varTheta +\delta +\omega \Big ) \end{pmatrix}, \\ & J_{23}=\begin{pmatrix} -\frac{nP(n)\beta ^V(n)V_n}{\left\langle k\right\rangle }& \cdots & -\frac{mP(m)\beta ^V(n)V_n}{\left\langle k\right\rangle }\\ \vdots & \vdots & \vdots \\ -\frac{nP(n)\beta ^V(m)V_m}{\left\langle k\right\rangle }& \cdots & -\frac{mP(m)\beta ^V(m)V_m}{\left\langle k\right\rangle } \end{pmatrix},\\ & J_{11}=\begin{pmatrix} -\Big (\beta ^S(n) \varTheta +x_n+\omega \Big )& \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & -\Big (\beta ^S(m) \varTheta +x_m+\omega \Big ) \end{pmatrix}, \\ & J_{33}=\begin{pmatrix} \frac{nP(n)\Big (\beta ^S(n) S_n+\beta ^V(n)V_n\Big )}{\left\langle k\right\rangle }-(\gamma +\omega )& \cdots & \frac{mP(m)\Big (\beta ^S(n) S_n+\beta ^V(n)V_n\Big )}{\left\langle k\right\rangle }\\ \vdots & \ddots & \vdots \\ \frac{nP(n)\Big (\beta ^S(m) S_m+\beta ^V(m)V_m\Big )}{\left\langle k\right\rangle }& \cdots & \frac{mP(m)\Big (\beta ^S(m) S_m+\beta ^V(m)V_m\Big )}{\left\langle k\right\rangle }-(\gamma +\omega ) \end{pmatrix}, \end{aligned}$$

$$\begin{aligned} & J_{31}=\begin{pmatrix} \beta ^S(n) \varTheta & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & \beta ^S(m) \varTheta \end{pmatrix}, \\ & J_{32}=\begin{pmatrix} \beta ^V(n)\varTheta & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & \beta ^V(m)\varTheta \end{pmatrix}, & J_{43}=\begin{pmatrix} \gamma & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & \gamma \end{pmatrix}, J_{44}=\begin{pmatrix} -\omega & \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & -\omega \end{pmatrix}, \\ & J_{53}=\begin{pmatrix} f_n& \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & f_m \end{pmatrix}, J_{55}=\begin{pmatrix} g_n& \cdots & 0\\ \vdots & \ddots & \vdots \\ 0& \cdots & g_m \end{pmatrix}, \\ g_n= & \xi (1-2x_n)W_n, \\ f_n= & \xi x_n(1-x_n)\Big ((1-\sigma )\beta ^V(n)\\ & +(\sigma \delta -1)\beta ^S(n)\Big )C_I(1-e^{-\alpha \tau }). \end{aligned}$$

Then the characteristic equation of $E_0$ for J can be inferred

$$\begin{aligned} \left| \begin{array}{ccccc} J_{11}-\lambda {{\textbf {I}}}& J_{12}& J_{13}& {{\textbf {0}}}& J_{15}\\ {{\textbf {0}}}& J_{22}-\lambda {{\textbf {I}}}& {{\textbf {0}}}& {{\textbf {0}}}& J_{25}\\ {{\textbf {0}}}& {{\textbf {0}}}& J_{33}-\lambda {{\textbf {I}}}& {{\textbf {0}}}& {{\textbf {0}}}\\ {{\textbf {0}}}& {{\textbf {0}}}& J_{43}& J_{44}-\lambda {{\textbf {I}}}& {{\textbf {0}}}\\ {{\textbf {0}}}& {{\textbf {0}}}& {{\textbf {0}}}& {{\textbf {0}}}& J_{55}-\lambda {{\textbf {I}}} \end{array}\right| _{E_0}= & 0, \end{aligned}$$

(16)

where ${{\textbf {I}}}$ is an identity matrix with appropriate dimension. The characteristic equation (16) can be further transformed into

$$\begin{aligned} & (\lambda +\gamma +\omega )^{\mathbb {n}-1}\Big (\lambda -\frac{\sum _{k=n}^{m}kP(k)\beta ^S(k) }{\left\langle k\right\rangle }+\gamma +\omega \Big )\nonumber \\ & \times \Big ((\lambda +\omega )^2(\lambda +\omega +\delta )(\lambda +\xi C_V(1-e^{-\alpha \tau }))\Big )^\mathbb {n}=0.\nonumber \\ \end{aligned}$$

(17)

Clearly, when ${\mathscr {R}}_0<1$, $E_0$ is locally asymptotically stable.

Similarly, the characteristic equation of $E_1$ is

$$\begin{aligned} \left| \begin{array}{ccccc} J_{11}-\lambda {{\textbf {I}}}& J_{12}& J_{13}& {{\textbf {0}}}& J_{15}\\ J_{21}& J_{22}-\lambda {{\textbf {I}}}& J_{23}& {{\textbf {0}}}& J_{25}\\ {{\textbf {0}}}& {{\textbf {0}}}& J_{33}-\lambda {{\textbf {I}}}& {{\textbf {0}}}& {{\textbf {0}}}\\ {{\textbf {0}}}& {{\textbf {0}}}& J_{43}& J_{44}-\lambda {{\textbf {I}}}& {{\textbf {0}}}\\ {{\textbf {0}}}& {{\textbf {0}}}& {{\textbf {0}}}& {{\textbf {0}}}& J_{55}-\lambda {{\textbf {I}}} \end{array}\right| _{E_1}= & 0.\nonumber \\ \end{aligned}$$

(18)

By calculation, the characteristic equation (18) must have a characteristic root $\lambda =\xi C_V(1-e^{-\alpha \tau })>0$. Therefore, $E_1$ is always unstable.

When ${\mathscr {R}}_0>1$, there must be a positive eigenvalue $\lambda$ of ${{\textbf {J}}}$. Based on the Perron-Frobenius theorem [52], the largest real part of all eigenvalues of ${{\textbf {J}}}$ is positive only when ${\mathscr {R}}_0>1$. Therefore, it is $\lim _{t\rightarrow \infty }\inf I(t)\ge \varepsilon$ based on the theorem in [53]. This completes the proof.

B Proof of Theorem 3

In the light of Theorem 2, there is a constant $0<\zeta <\frac{1}{3}$ and a large enough constant $T>0$ such that $I_k(t)\ge \zeta$ holds for all $t>T$. Based on the first equation of system (9), it yields

$$\begin{aligned} {\dot{S}}_k\le & \omega -\beta ^S(k) \zeta S_k-x_kS_k+\delta \Big (1-S_k-I_k-R_k\Big )-\omega S_k \\\le & \omega +\delta -\Big (\beta ^S(k) \zeta +x_k+\omega +\delta \Big )S_k, t>T. \end{aligned}$$

In accordance with the standard comparison theorem of differential equation theory, for any given normal number $0<\zeta _1<\frac{\beta ^S(k) \zeta +x_k}{2(\beta ^S(k) \zeta +x_k+\omega +\delta )}$, $\exists t_1>T$, for $t>t_1$, $S_k(t)\le {\mathbb {X}}_k^{(1)}-\zeta _1$, where

$$\begin{aligned} {\mathbb {X}}_k^{(1)}=\frac{\omega +\delta }{\beta ^S(k) \zeta +x_k+\omega +\delta }+2\zeta _1<1. \end{aligned}$$

Similarly,

$$\begin{aligned} {\dot{V}}_k\le & x_kS_k-\beta ^V(k)\zeta V_k-(\omega +\delta ) V_k\\\le & x_k-\Big (\beta ^V(k)\zeta +x_k+\omega +\delta \Big )V_k, t>t_1. \end{aligned}$$

Therefore, for any given constant $0<\zeta _2<\min \Big \{\frac{1}{2}, \zeta _1,$ $\frac{\beta ^V(k)\zeta +\omega +\delta }{2(\beta ^V(k)\zeta +x_k+\omega +\delta )}\Big \}$, $\exists t_2>t_1$, when $t>t_2$, $V_k(t)\le {\mathbb {Y}}_k^{(1)}-\zeta _2$, where

$$\begin{aligned} {\mathbb {Y}}_k^{(1)}=\frac{x_k}{\beta ^V(k)\zeta +x_k+\omega +\delta }+2\zeta _2<1. \end{aligned}$$

Utilizing the third equation of system (9) yields

$$\begin{aligned} {\dot{I}}_k\le & \beta ^S(k) (1-I_k)+\Big (\beta ^V(k)-\beta ^S(k) \Big )V_k-(\gamma +\omega )I_k \\\le & \beta ^S(k) -\Big (\beta ^S(k) +\gamma +\omega \Big )I_k, t>t_2. \end{aligned}$$

Thus, for an arbitrary given constant $0<\zeta _3<\min \Big \{\frac{1}{3}, \zeta _2,$ $\frac{\gamma +\omega }{2(\beta ^S(k) +\gamma +\omega )}\Big \}$, $\exists t_3>t_2$, when $t>t_3$, $I_k(t)\le {\mathbb {Z}}_k^{(1)}-\zeta _3$, where

$$\begin{aligned} {\mathbb {Z}}_k^{(1)}=\frac{\beta ^S(k) }{\beta ^S(k) +\gamma +\omega }+2\zeta _3<1. \end{aligned}$$

With the help of the fourth equation of system (9), it is straightforward to obtain

$$\begin{aligned} {\dot{R}}_k\le \gamma -(\gamma +\omega ) R_k, t>t_3. \end{aligned}$$

As a consequence, for a given positive constant $0<\zeta _4<\min \Big \{\frac{1}{4}, \zeta _3, \frac{\omega }{2(\gamma +\omega )}\Big \}$, $\exists t_4>t_3$, and for $t>t_4$, $R_k(t)\le {\mathbb {M}}_k^{(1)}-\zeta _4$, where

$$\begin{aligned} {\mathbb {M}}_k^{(1)}=\frac{\gamma }{\gamma +\omega }+2\zeta _4<1. \end{aligned}$$

Next, the first equation of substituting $V_k(t)\le {\mathbb {Y}}_k^{(1)}-\zeta _2, I_k(t)\le {\mathbb {Z}}_k^{(1)}-\zeta _3, R_k(t)\le {\mathbb {M}}_k^{(1)}-\zeta _4$ into system (9) is

$$\begin{aligned} {\dot{S}}_k\ge & \omega -\beta ^S(k) S_k-x_kS_k+\delta \Big (1-S_k-I_k-R_k\Big )-\omega S_k \\\ge & \omega +\delta \Big (1-{\mathbb {Z}}_k^{(1)}-{\mathbb {M}}_k^{(1)}\Big )-\Big (\beta ^S(k) +x_k\\ & +\omega +\delta \Big )S_k, t>t_4. \end{aligned}$$

Consequently, for

$$0<\zeta _5<\min \Big \{\frac{1}{5}, \zeta _4, \frac{\omega +\delta (1-{\mathbb {Z}}_k^{(1)}-{\mathbb {M}}_k^{(1)})}{2(\beta ^S(k) +x_k+\omega +\delta )}\Big \},$$

$\exists t_5>t_4$, when $t>t_5$, $S_k(t)\ge \mathbb {x}_k^{(1)}+\zeta _5$, where

$$\begin{aligned} \mathbb {x}_k^{(1)}=\frac{\omega +\delta \Big (1-{\mathbb {Z}}_k^{(1)}-{\mathbb {M}}_k^{(1)}\Big )}{\beta ^S(k) +x_k+\omega +\delta }-2\zeta _5>0. \end{aligned}$$

Similarly,

$$\begin{aligned} {\dot{V}}_k\ge & x_k\Big (1-V_k-I_k-R_k\Big )-\beta ^V(k) V_k-(\omega +\delta ) V_k\\\ge & x_k\Big (1-{\mathbb {Z}}_k^{(1)}-{\mathbb {M}}_k^{(1)}\Big )-\Big (\beta ^V(k)+x_k\\ & +\omega +\delta \Big )V_k, t>t_5. \end{aligned}$$

Thereby, for $0<\zeta _6<\min \Big \{\frac{1}{6}, \zeta _5, \frac{x_k(1-{\mathbb {Z}}_k^{(1)}-{\mathbb {M}}_k^{(1)})}{2(\beta ^V(k)+x_k+\omega +\delta )}\Big \}$, $\exists t_6>t_5$, for $t>t_6$, $V_k(t)\ge \mathbb {y}_k^{(1)}+\zeta _6$, where

$$\begin{aligned} \mathbb {y}_k^{(1)}=\frac{x_k\Big (1-{\mathbb {Z}}_k^{(1)}-{\mathbb {M}}_k^{(1)}\Big )}{\beta ^V(k)+x_k+\omega +\delta }-2\zeta _6>0. \end{aligned}$$

Furthermore,

$$\begin{aligned} {\dot{I}}_k\ge & \beta ^S(k) \zeta \Big (1-I_k-R_k\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big )\zeta V_k\\ & -(\gamma +\omega )I_k \\\ge & \beta ^S(k) \zeta \Big (1-{\mathbb {M}}_k^{(1)}\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big )\zeta {\mathbb {Y}}_k^{(1)}\\ & -\Big (\beta ^S(k) \zeta +\gamma +\omega \Big )I_k, t>t_6. \end{aligned}$$

As a consequence, for

$$0<\zeta _7<\min \Big \{\frac{1}{7}, \zeta _6,$$

$\frac{\beta ^S(k) \zeta (1-{\mathbb {M}}_k^{(1)})-(\beta ^S(k) -\beta ^V(k))\zeta {\mathbb {Y}}_k^{(1)}}{2(\beta ^S(k) \zeta +\gamma +\omega )}\Big \}$, $\exists t_7>t_6$, when $t>t_7$, $I_k(t)\ge \mathbb {z}_k^{(1)}+\zeta _7$, where

$$\begin{aligned} \mathbb {z}_k^{(1)}= & \frac{\beta ^S(k) \zeta \Big (1-{\mathbb {M}}_k^{(1)}\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big )\zeta {\mathbb {Y}}_k^{(1)}}{\beta ^S(k) \zeta +\gamma +\omega }\\ & -2\zeta _7>0. \end{aligned}$$

In addition,

$$\begin{aligned} {\dot{R}}_k\ge \gamma \Big (1-{\mathbb {X}}_k^{(1)}-{\mathbb {Y}}_k^{(1)}\Big )-(\gamma +\omega ) R_k, t>t_7. \end{aligned}$$

Hence, for $0<\zeta _8<\min \Big \{\frac{1}{8}, \zeta _7, \frac{\gamma (1-{\mathbb {X}}_k^{(1)}-{\mathbb {Y}}_k^{(1)})}{2(\gamma +\omega )}\Big \}$, $\exists t_8>t_7$, for $t>t_8$, $R_k(t)\ge \mathbb {m}_k^{(1)}+\zeta _8$, where

$$\begin{aligned} \mathbb {m}_k^{(1)}=\frac{\gamma \Big (1-{\mathbb {X}}_k^{(1)}-{\mathbb {Y}}_k^{(1)}\Big )}{\gamma +\omega }-2\zeta _8>0. \end{aligned}$$

Due to the fact that $\zeta$ is a small positive constant, then $0<\mathbb {x}_k^{(1)}<S_k<{\mathbb {X}}_k^{(1)}<1, 0<\mathbb {y}_k^{(1)}<V_k<{\mathbb {Y}}_k^{(1)}<1, 0<\mathbb {z}_k^{(1)}<I_k<{\mathbb {Z}}_k^{(1)}<1$ and $0<\mathbb {m}_k^{(1)}<R_k<{\mathbb {M}}_k^{(1)}<1$. Let

$$\begin{aligned} \varPhi ^{(i)}= & \frac{1}{\left\langle k\right\rangle }\sum _{k=n}^{m}kP(k){\mathbb {Z}}_k^{(i)},\nonumber \\ \phi ^{(i)}= & \frac{1}{\left\langle k\right\rangle }\sum _{k=n}^{m}kP(k)\mathbb {z}_k^{(i)}, i=1,2,\dots . \end{aligned}$$

(19)

Hence, $0<\phi ^{(1)}\le \varTheta (t)\le \varPhi ^{(1)}<1, t>t_8$.

Substituting $V_k(t)\ge \mathbb {y}_k^{(1)}+\zeta _6, I_k(t)\ge \mathbb {z}_k^{(1)}+\zeta _7, R_k(t)\le \mathbb {m}_k^{(1)} +\zeta _8$ into the first equation of system (9), it has

$$\begin{aligned} {\dot{S}}_k\le & \omega -\beta ^S(k) \phi ^{(1)} S_k-x_kS_k+\delta \Big (1-S_k-I_k-R_k\Big )\\ & -\omega S_k \\\le & \omega +\delta \Big (1-\mathbb {z}_k^{(1)}-\mathbb {m}_k^{(1)}\Big )-\Big (\beta ^S(k) \phi ^{(1)}+x_k\\ & +\omega +\delta \Big )S_k, t>t_8. \end{aligned}$$

Therefore, for $0<\zeta _9<\min \{\frac{1}{9}, \zeta _8\}$, $\exists t_9>t_8$, it obtains

$$\begin{aligned} S_k(t)\le & {\mathbb {X}}_k^{(2)}\triangleq \min \Big \{{\mathbb {X}}_k^{(1)}\\ & -\zeta _1, \frac{\omega +\delta \Big (1-\mathbb {z}_k^{(1)}-\mathbb {m}_k^{(1)}\Big )}{\beta ^S(k) \phi ^{(1)}+x_k+\omega +\delta }\Big \}, t>t_9. \end{aligned}$$

Similarly,

$$\begin{aligned} {\dot{V}}_k\le & x_kS_k-\beta ^V(k)\phi ^{(1)} V_k-(\omega +\delta ) V_k\\\le & x_k\Big (1-\mathbb {z}_k^{(1)}-\mathbb {m}_k^{(1)}\Big )-\Big (\beta ^V(k)\phi ^{(1)}+x_k\\ & +\omega +\delta \Big )V_k, t>t_9. \end{aligned}$$

As a consequence, for $0<\zeta _{10}<\min \{\frac{1}{10}, \zeta _9\}$, $\exists t_{10}>t_9$, one gets

$$\begin{aligned} V_k(t)\le & {\mathbb {Y}}_k^{(2)}\triangleq \min \Big \{{\mathbb {Y}}_k^{(1)}\\ & -\zeta _2,\frac{x_k\Big (1-\mathbb {z}_k^{(1)}-\mathbb {m}_k^{(1)}\Big )}{\beta ^V(k)\phi ^{(1)}+x_k+\omega +\delta }\Big \}, t>t_{10}. \end{aligned}$$

In addition,

$$\begin{aligned} {\dot{I}}_k\le & \Big (\beta ^S(k) S_k+\beta ^V(k)V_k\Big )\varPhi ^{(1)}-(\gamma +\omega ) I_k \\\le & \Big (\beta ^S(k) \Big (1-\mathbb {m}_k^{(1)}\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big ) \mathbb {y}_k^{(1)}\Big )\varPhi ^{(1)}\\ & -\Big (\beta ^S(k) \varPhi ^{(1)}+\gamma +\omega \Big )I_k, t>t_{10}. \end{aligned}$$

Hence, for $0<\zeta _{11}<\min \{\frac{1}{11}, \zeta _{10}\}$, $\exists t_{11}>t_{10}$, one has

$$\begin{aligned} & I_k(t)\le {\mathbb {Z}}_k^{(2)}\triangleq \min \Big \{{\mathbb {Z}}_k^{(1)}\\ & -\zeta _3,\frac{\Big (\beta ^S(k) \Big (1-\mathbb {m}_k^{(1)}\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big ) \mathbb {y}_k^{(1)}\Big )\varPhi ^{(1)}}{\beta ^S(k) \varPhi ^{(1)}+\gamma +\omega }\Big \},\\ & \quad t>t_{11}. \end{aligned}$$

Furthermore,

$$\begin{aligned} {\dot{R}}_k\le \gamma \Big (1-\mathbb {x}_k^{(1)}-\mathbb {y}_k^{(1)}\Big )-(\gamma +\omega ) R_k, t>t_{11}. \end{aligned}$$

Consequently, for $0<\zeta _{12}<\min \{\frac{1}{12}, \zeta _{11}\}$, $\exists t_{12}>t_{11}$, it has

$$\begin{aligned} R_k(t)\le & {\mathbb {M}}_k^{(2)}\triangleq \min \Big \{{\mathbb {M}}_k^{(1)}\\ & -\zeta _4,\frac{\gamma \Big (1-\mathbb {x}_k^{(1)}-\mathbb {y}_k^{(1)}\Big )}{\gamma +\omega }\Big \}, t>t_{12}. \end{aligned}$$

The first equation substituting $V_k(t)\le {\mathbb {Y}}_k^{(2)}, I_k(t)\le {\mathbb {Z}}_k^{(2)},R_k(t)\le {\mathbb {M}}_k^{(2)}$ into system (9) gives

$$\begin{aligned} {\dot{S}}_k\ge & \omega -\beta ^S(k) \varPhi ^{(1)} S_k-x_kS_k+\delta \Big (1-S_k-I_k-R_k\Big )\\ & -\omega S_k \\\ge & \omega +\delta \Big (1-{\mathbb {Z}}_k^{(2)}-{\mathbb {M}}_k^{(2)}\Big )-\Big (\beta ^S(k) \varPhi ^{(1)}+x_k\\ & +\omega +\delta \Big )S_k, t>t_{12}. \end{aligned}$$

Therefore, for

$$0<\zeta _{13}<\min \Big \{\frac{1}{13}, \zeta _{12}, \frac{\omega +\delta (1-{\mathbb {Z}}_k^{(2)}-{\mathbb {M}}_k^{(2)})}{2(\beta ^S(k) \varPhi ^{(1)}+x_k+\omega +\delta )}\Big \},$$

$\exists t_{13}>t_{12},$ when $t>t_{13}$, $S_k(t)\ge \mathbb {x}_k^{(2)}+\zeta _{13}$, where

$$\begin{aligned} \mathbb {x}_k^{(2)}= \max \Big \{\mathbb {x}_k^{(1)}+\zeta _5,\frac{\omega +\delta \Big (1-{\mathbb {Z}}_k^{(2)}-{\mathbb {M}}_k^{(2)}\Big )}{\beta ^S(k) \varPhi ^{(1)}+x_k+\omega +\delta }-2\zeta _{13}\Big \}. \end{aligned}$$

Similarly,

$$\begin{aligned} {\dot{V}}_k\ge & x_k\Big (1-V_k-I_k-R_k\Big )-\beta ^V(k)\varPhi ^{(1)} V_k-(\omega +\delta ) V_k\\\ge & x_k\Big (1-{\mathbb {Z}}_k^{(2)}-{\mathbb {M}}_k^{(2)}\Big )-\Big (\beta ^V(k)\varPhi ^{(1)}+x_k\\ & +\omega +\delta \Big )V_k, t>t_{13}. \end{aligned}$$

Thereby, for

$$0<\zeta _{14}<\min \Big \{\frac{1}{14}, \zeta _{13}, \frac{x_k(1-{\mathbb {Z}}_k^{(2)}-{\mathbb {M}}_k^{(2)})}{2(\beta ^V(k)\varPhi ^{(1)}+x_k+\omega +\delta )}\Big \},$$

$\exists t_{14}>t_{13}$, for $t>t_{14}$, $V_k(t)\ge \mathbb {y}_k^{(2)}+\zeta _{14}$, where

$$\begin{aligned} \mathbb {y}_k^{(2)}=\max \Big \{\mathbb {y}_k^{(1)}+\zeta _6,\frac{x_k\Big (1-{\mathbb {Z}}_k^{(2)}-{\mathbb {M}}_k^{(2)}\Big )}{\beta ^V(k)\varPhi ^{(1)}+x_k+\omega +\delta }-2\zeta _{14}\Big \}. \end{aligned}$$

In addition,

$$\begin{aligned} {\dot{I}}_k\ge & \Big (\beta ^S(k) \Big (1-V_k-I_k-R_k\Big )+\beta ^V(k) V_k\Big )\phi ^{(1)}\\ & -(\gamma +\omega ) I_k \\\ge & \Big (\beta ^S(k) \Big (1-{\mathbb {M}}_k^{(2)}\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big ) {\mathbb {Y}}_k^{(2)}\Big )\phi ^{(1)}\\ & -\Big (\beta ^S(k) \phi ^{(1)}+\gamma +\omega \Big )I_k, t>t_{14}. \end{aligned}$$

As a consequence, for $0<\zeta _{15}<\min \Big \{\frac{1}{15}, \zeta _{14},$

$\frac{(\beta ^S(k) (1-{\mathbb {M}}_k^{(2)})-(\beta ^S(k) -\beta ^V(k)) {\mathbb {Y}}_k^{(2)})\phi ^{(1)}}{2(\beta ^S(k) \zeta +\gamma +\omega )}\Big \}$, $\exists t_{15}>t_{14}$, for $t>t_{15}$, $I_k(t)\ge \mathbb {z}_k^{(2)}+\zeta _{15}$, where

$$\begin{aligned} & \mathbb {z}_k^{(2)}=\max \Big \{\mathbb {z}_k^{(1)}+\zeta _7,\\ & \frac{\Big (\beta ^S(k) \Big (1-{\mathbb {M}}_k^{(2)}\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big ) {\mathbb {Y}}_k^{(2)}\Big )\phi ^{(1)}}{\beta ^S(k) \phi ^{(1)}+\gamma +\omega }-2\zeta _{15}\Big \}. \end{aligned}$$

Furthermore,

$$\begin{aligned} {\dot{R}}_k\ge \gamma \Big (1-{\mathbb {X}}_k^{(2)}-{\mathbb {Y}}_k^{(2)}\Big )-(\gamma +\omega ) R_k, t>t_{15}. \end{aligned}$$

Hence, for $0<\zeta _{16}<\min \Big \{\frac{1}{16}, \zeta _{15}, \frac{\gamma (1-{\mathbb {X}}_k^{(2)}-{\mathbb {Y}}_k^{(2)})}{2(\gamma +\omega )}\Big \}$, $\exists t_{16}>t_{15}$, when $t>t_{16}$, $R_k(t)\ge \mathbb {m}_k^{(2)}+\zeta _{16}$, where

$$\begin{aligned} \mathbb {m}_k^{(2)}=\max \Big \{\mathbb {m}_k^{(1)}+\zeta _7, \frac{\gamma \Big (1-{\mathbb {X}}_k^{(2)}-{\mathbb {Y}}_k^{(2)}\Big )}{\gamma +\omega }-2\zeta _{16}\Big \}. \end{aligned}$$

Consequently, $0<\phi ^{(2)}\le \varTheta (t)\le \varPhi ^{(2)}, t>t_{16}$.

It is similarly possible to perform the $\ell$-th step and obtain 8 sequences: $\big \{{\mathbb {X}}_k^{(\ell )}\big \}, \big \{{\mathbb {Y}}_k^{(\ell )}\big \}$, $\big \{{\mathbb {Z}}_k^{(\ell )}\big \}$, $\big \{{\mathbb {M}}_k^{(\ell )}\big \}$, $\big \{\mathbb {x}_k^{(\ell )}\big \}, \big \{\mathbb {y}_k^{(\ell )}\big \}$, $\big \{\mathbb {z}_k^{(\ell )}\big \}$ and $\big \{\mathbb {m}_k^{(\ell )}\big \}$. Due to the fact that the first four are monotonically increasing and the last four are strictly decreasing, there is a large positive integer N, when $\ell >N$,

$$\begin{aligned} \begin{array}{l} {\mathbb {X}}_k^{(\ell )}=\frac{\omega +\delta \Big (1-\mathbb {z}_k^{(\ell -1)}-\mathbb {m}_k^{(\ell -1)}\Big )}{\beta ^S(k) \phi ^{(\ell -1)}+x_k+\omega +\delta }+\zeta _{8\ell -7},\\ {\mathbb {Y}}_k^{(\ell )}=\frac{x_k\Big (1-\mathbb {z}_k^{(\ell -1)}-\mathbb {m}_k^{(\ell -1)}\Big )}{\beta ^V(k)\phi ^{(\ell -1)}+x_k+\omega +\delta }+\zeta _{8\ell -6},\\ {\mathbb {Z}}_k^{(\ell )}=\frac{\Big (\beta ^S(k) \Big (1-\mathbb {m}_k^{(\ell -1)}\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big ) \mathbb {y}_k^{(\ell -1)}\Big )\varPhi ^{(\ell -1)}}{\beta ^S(k) \varPhi ^{(\ell -1)}+\gamma +\omega }\\ \hspace{1.2cm}+\zeta _{8\ell -5},\\ {\mathbb {M}}_k^{(\ell )}=\frac{\gamma \Big (1-\mathbb {x}_k^{(\ell -1)}-\mathbb {y}_k^{(\ell -1)}\Big )}{\gamma +\omega }+\zeta _{8\ell -4},\\ \mathbb {x}_k^{(\ell )}=\frac{\omega +\delta \Big (1-{\mathbb {Z}}_k^{(\ell )}-{\mathbb {M}}_k^{(\ell )}\Big )}{\beta ^S(k) \varPhi ^{(\ell -1)}+x_k+\omega +\delta }-2\zeta _{8\ell -3},\\ \mathbb {y}_k^{(\ell )}=\frac{x_k\Big (1-{\mathbb {Z}}_k^{(\ell )}-{\mathbb {M}}_k^{(\ell )}\Big )}{\beta ^V(k)\varPhi ^{(\ell -1)}+x_k+\omega +\delta }-2\zeta _{8\ell -2},\\ \mathbb {z}_k^{(\ell )}=\frac{\Big (\beta ^S(k) \Big (1-{\mathbb {M}}_k^{(\ell )}\Big )-\Big (\beta ^S(k) -\beta ^V(k)\Big ) {\mathbb {Y}}_k^{(\ell )}\Big )\phi ^{(\ell -1)}}{\beta ^S(k) \phi ^{(\ell -1)}+\gamma +\omega }\\ \hspace{1.2cm}-2\zeta _{8\ell -1},\\ \mathbb {m}_k^{(\ell )}=\frac{\gamma \Big (1-{\mathbb {X}}_k^{(\ell )}-{\mathbb {Y}}_k^{(\ell )}\Big )}{\gamma +\omega }-2\zeta _{8\ell }. \end{array} \end{aligned}$$

(20)

Then,

$$\begin{aligned} \mathbb {x}_k^{(\ell )}\le & S_k(t)\le {\mathbb {X}}_k^{(\ell )}, \mathbb {y}_k^{(\ell )}\le V_k(t)\le {\mathbb {Y}}_k^{(\ell )},\end{aligned}$$

(21)

$$\begin{aligned} \mathbb {z}_k^{(\ell )}\le & I_k(t)\le {\mathbb {Z}}_k^{(\ell )}, \mathbb {m}_k^{(\ell )}\le R_k(t)\le {\mathbb {M}}_k^{(\ell )}, t>t_{8\ell }.\nonumber \\ \end{aligned}$$

(22)

Observe that $0<\zeta _{\ell }<\frac{1}{\ell }$, then $\zeta _{\ell }\rightarrow 0$ when $\ell \rightarrow \infty$. For the ten sequences of (19) and (20), when $i\rightarrow \infty$ and $\ell \rightarrow \infty$, it has

$$\begin{aligned} & \lim _{i\rightarrow \infty }\varPhi ^{(i)}=\frac{1}{\left\langle k\right\rangle }\sum _{k=n}^{m}kP(k){\mathbb {Z}}_k\triangleq \varPhi ,\nonumber \\ & \lim _{i\rightarrow \infty }\phi ^{(i)}=\frac{1}{\left\langle k\right\rangle }\sum _{k=n}^{m}kP(k)\mathbb {z}_k\triangleq \phi . \end{aligned}$$

(23)

$$\begin{aligned} & \lim \limits _{\ell \rightarrow \infty }{\mathbb {X}}_k^{(\ell )}\nonumber \\ & =\frac{\omega \Big (\beta ^V(k)\varPhi +\omega +\delta \Big )}{\Big (\beta ^S(k) \varPhi +\omega +x_k\Big )\Big (\beta ^V(k)\varPhi +\omega +\delta \Big )-x_k\delta }\triangleq {\mathbb {X}}_k,\nonumber \\ & \lim \limits _{\ell \rightarrow \infty }{\mathbb {Y}}_k^{(\ell )}\nonumber \\ & =\frac{\omega x_k}{\Big (\beta ^S(k) \varPhi +\omega +x_k\Big )\Big (\beta ^V(k)\varPhi +\omega +\delta \Big )-x_k\delta }\triangleq {\mathbb {Y}}_k,\nonumber \\ & \lim \limits _{\ell \rightarrow \infty }{\mathbb {Z}}_k^{(\ell )}\nonumber \\ & =\frac{\omega \Big (\beta ^S(k) \Big (\beta ^V(k)\varPhi +\omega +\delta \Big )+x_k\beta ^V(k)\Big )\varPhi }{(\gamma +\omega )\Big (\Big (\beta ^S(k) \varPhi +\omega +x_k\Big )\Big (\beta ^V(k)\varPhi +\omega +\delta \Big )-x_k\delta \Big )}\triangleq {\mathbb {Z}}_k,\nonumber \\ & \lim \limits _{\ell \rightarrow \infty }{\mathbb {M}}_k^{(\ell )}\nonumber \\ & =\frac{\gamma \Big (\beta ^S(k) \Big (\beta ^V(k)\varPhi +\omega +\delta \Big )+\beta ^V(k)\Big )\varPhi }{(\gamma +\omega )\Big (\Big (\beta ^S(k) \varPhi +\omega +x_k\Big )\Big (\beta ^V(k)\varPhi +\omega +\delta \Big )-x_k\delta \Big )}\triangleq {\mathbb {M}}_k,\nonumber \\ & \lim \limits _{\ell \rightarrow \infty }\mathbb {x}_k^{(\ell )}=\frac{\omega \Big (\beta ^V(k)\phi +\omega +\delta \Big )}{\Big (\beta ^S(k) \phi +\omega +x_k\Big )\Big (\beta ^V(k)\phi +\omega +\delta \Big )-x_k\delta }\triangleq \mathbb {x}_k,\nonumber \\ & \lim \limits _{\ell \rightarrow \infty }\mathbb {y}_k^{(\ell )}\nonumber \\ & =\frac{\omega x_k}{\Big (\beta ^S(k) \phi +\omega +x_k\Big )\Big (\beta ^V(k)\phi +\omega +\delta \Big )-x_k\delta }\triangleq \mathbb {y}_k,\nonumber \\ & \lim \limits _{\ell \rightarrow \infty }\mathbb {z}_k^{(\ell )}\nonumber \\ & =\frac{\omega \Big (\beta ^S(k) \Big (\beta ^V(k)\phi +\omega +\delta \Big )+x_k\beta ^V(k)\Big )\phi }{(\gamma +\omega )\Big (\Big (\beta ^S(k) \phi +\omega +x_k\Big )\Big (\beta ^V(k)\phi +\omega +\delta \Big )-x_k\delta \Big )}\triangleq \mathbb {z}_k,\nonumber \\ & \lim \limits _{\ell \rightarrow \infty }\mathbb {m}_k^{(\ell )}\nonumber \\ & =\frac{\gamma \Big (\beta ^S(k) \Big (\beta ^V(k)\phi +\omega +\delta \Big )+\beta ^V(k)\Big )\phi }{(\gamma +\omega )\Big (\Big (\beta ^S(k) \phi +\omega +x_k\Big )\Big (\beta ^V(k)\phi +\omega +\delta \Big )-x_k\delta \Big )}\triangleq \mathbb {m}_k.\nonumber \\ \end{aligned}$$

(24)

Substituting (24) into (23)

$$\begin{aligned} \begin{array}{l} \frac{1}{\left\langle k\right\rangle }\sum \limits _{k=n}^{m}\frac{kP(k)\omega \Big (\beta ^S(k) \Big (\beta ^V(k)\varPhi +\omega +\delta \Big )+x_k\beta ^V(k)\Big )}{(\gamma +\omega )\Big (\Big (\beta ^S(k) \varPhi +\omega +x_k\Big )\Big (\beta ^V(k)\varPhi +\omega +\delta \Big )-x_k\delta \Big )}=1,\\ \end{array} \\ \begin{array}{l} \frac{1}{\left\langle k\right\rangle }\sum \limits _{k=n}^{m}\frac{kP(k)\omega \Big (\beta ^S(k) \Big (\beta ^V(k)\phi +\omega +\delta \Big )+x_k\beta ^V(k)\Big )}{(\gamma +\omega )\Big (\Big (\beta ^S(k) \phi +\omega +x_k\Big )\Big (\beta ^V(k)\phi +\omega +\delta \Big )-x_k\delta \Big )}=1. \end{array} \end{aligned}$$

Subtracting the above two equations, one gets

$$\begin{aligned} \frac{1}{\left\langle k\right\rangle }\sum _{k=n}^{m}\frac{kP(k)\varXi _1(k)(\varPhi -\phi )}{\varXi _2(k)}=0, \end{aligned}$$

where

$$\begin{aligned} \varXi _1(k)= & \Big (\beta ^S(k) (\omega +\delta )+x_k\beta ^V(k)\Big )\beta ^S(k) \beta ^V(k)(\varPhi +\phi )\\ & +\Big (\beta ^S(k) (\omega +\delta )+x_k\beta ^V(k)\Big )\Big (\beta ^S(k) (\omega +\delta )\\ & +\beta ^V(k)(\omega +x_k)\Big )+\Big (\beta ^S(k) \beta ^V(k)\Big )^2\varPhi \phi \\ & +\beta ^S(k) \beta ^V(k)(\omega ^2+\omega x_k+\omega \delta ),\\ \varXi _2(k)= & \Big ((\gamma +\omega )\Big (\Big (\beta ^S(k) \varPhi +\omega +x_k\Big )\Big (\beta ^V(k)\varPhi +\omega +\delta \Big )\\ & -x_k\delta \Big )\Big )\Big ((\gamma +\omega )\Big (\Big (\beta ^S(k) \phi +\omega +x_k\Big )\\ & \times \Big (\beta ^V(k)\phi +\omega +\delta \Big )-x_k\delta \Big )\Big ). \end{aligned}$$

Therefore, $\varPhi =\phi$ which is equivalent to ${\mathbb {X}}_k=\mathbb {x}_k, {\mathbb {Y}}_k=\mathbb {y}_k, {\mathbb {Z}}_k=\mathbb {z}_k$ and ${\mathbb {M}}_k=\mathbb {m}_k$. Therefore, based on (21), it gives

$$\begin{aligned} \lim _{t\rightarrow \infty }S_k(t)= & {\mathbb {X}}_k=\mathbb {x}_k,\quad \lim _{t\rightarrow \infty }V_k(t)={\mathbb {Y}}_k=\mathbb {y}_k,\\ \lim _{t\rightarrow \infty }I_k(t)= & {\mathbb {Z}}_k=\mathbb {z}_k,\quad \lim _{t\rightarrow \infty }R_k(t)={\mathbb {M}}_k=\mathbb {m}_k. \end{aligned}$$

On the basis of (10), it yields ${\mathbb {X}}_k=S_k^*, {\mathbb {Y}}_k=V_k^*$, ${\mathbb {Z}}_k=I_k^*$ and ${\mathbb {M}}_k=R_k^*$. This completes the proof of the theorem.