A deterministic model for the transmission dynamics of SIRS-type malaria in hosts and SI in mosquito populations is proposed. The host population is differentiated between naive, primary, and secondary susceptible individuals. Primary and... more
A deterministic model for the transmission dynamics of SIRS-type malaria in hosts and SI in mosquito populations is proposed. The host population is differentiated between naive, primary, and secondary susceptible individuals. Primary and secondary infected individuals (and also recovered) are differentiated from each other according to their degree of infectiousness. The impact of changing the relative susceptibilities of primary and secondary (with respect to naive) susceptible individuals on the dynamics is investigated. Also, the impact of changing the relative infectiousness of secondary infected, primary, and secondary recovered individuals (with respect to primary infected) on the transmission dynamics of malaria is studied.
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This work aims mainly to study the impact of experiencing asymptomatic anthroponotic cutaneous leishmaniasis (ACL) infection on the overall dynamics and outcomes of the disease. Therefore, a deterministic model for the transmission... more
This work aims mainly to study the impact of experiencing asymptomatic anthroponotic cutaneous leishmaniasis (ACL) infection on the overall dynamics and outcomes of the disease. Therefore, a deterministic model for the transmission dynamics of ACL of type SEAIS in the human host and SI in sandfly populations is proposed and mathematically analyzed. The model is shown to be well-posed. Its equilibrium and stability analyses are shown. The equilibrium analysis shows that the model has an ACL-free equilibrium that is proven to be locally and globally asymptotically stable if and only if R0<1. In addition, the model has a unique ACL-endemic equilibrium that is shown to exist and be locally asymptotically stable if and only if R0>1. Numerical simulations are performed to show the asymptotic stability of these equilibriums. In addition, the effect of ignoring asymptomatic infections is studied and the analysis shows that ignoring the development of asymptomatic infections overestima...
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This work aims mainly to study the controllability of pertussis infection in the presence of waning and natural booster of pertussis immunity and to study their impact on the overall dynamics and disease outcomes. Therefore, an SIVRWS... more
This work aims mainly to study the controllability of pertussis infection in the presence of waning and natural booster of pertussis immunity and to study their impact on the overall dynamics and disease outcomes. Therefore, an SIVRWS (Susceptible-Infected-Vaccinated-Recovered-Waned-Susceptible) model for pertussis infection spread in a demographically stationary, homogeneous, and fully symmetric mixing population is introduced. The model has been mathematically analyzed, where both equilibrium and stability analyses have been established, and uniform persistence of the model has been shown. The conditions on model parameters that ensure effective control of the infection have been derived. The effects of the interplay between waning and boosting pertussis immunity by re-exposure to Bordetella pertussis and vaccination on the dynamics have been investigated. The analytical results have been numerically confirmed and explained. The analysis reveals that ignoring the natural booster o...
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The aim of this research is to determine the impact of needle exchange programs as motivation for intravenous drug users to seek treatment for addiction. A mathematical model of the dynamics of a population of drug users that incorporates... more
The aim of this research is to determine the impact of needle exchange programs as motivation for intravenous drug users to seek treatment for addiction. A mathematical model of the dynamics of a population of drug users that incorporates a needle exchange program is formulated. We define the basic addiction reproduction number for the proposed model and explore its role in the prevalence and control of needle-sharing drug addiction. Specifically, the local stability of the injection-addiction free and endemic equilibria are determined. Sensitivity analysis is conducted to determine the impact of perturbations of key parameters on the basic reproduction number and endemic level of the subpopulations. Results include a notable impact of the ineffectiveness of the needle exchange program on the proportion of the subpopulation in treatment. Furthermore, conditions necessary for an injection-addiction free population are determined.
Eating behaviors among a large population of children are studied as a dynamic process driven by nonlinear interactions in the sociocultural school environment. The impact of food association learning on diet dynamics, inspired by a pilot... more
Eating behaviors among a large population of children are studied as a dynamic process driven by nonlinear interactions in the sociocultural school environment. The impact of food association learning on diet dynamics, inspired by a pilot study conducted among Arizona children in Pre-Kindergarten to 8th grades, is used to build simple population-level learning models. Qualitatively, mathematical studies are used to highlight the possible ramifications of instruction, learning in nutrition, and health at the community level. Model results suggest that nutrition education programs at the population-level have minimal impact on improving eating behaviors, findings that agree with prior field studies. Hence, the incorporation of food association learning may be a better strategy for creating resilient communities of healthy and non-healthy eaters. A Ratatouille effect can be observed when food association learners become food preference learners, a potential sustainable behavioral chang...
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We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model... more
We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R 0 is smaller than the eradication effort for r > 1 + (μ/α) and equa...
Research Interests: Algorithms, Mathematical Biology, Biomathematics, Communicable Diseases, Biological Sciences, and 11 moreHumans, Mathematical Sciences, Animals, Steady state, Survival Rate, Basic Reproduction Number, Reproduction number, Communicable disease control, Epidemic Model, Rate Constant, and Compartmental Model
The SIS model of Hadeler and Castillo-Chavez [9] with a constant transfer rate of susceptibles into a partially protected state has been modified to take into account vaccination at birth. The model shows backward bifurcation (existence... more
The SIS model of Hadeler and Castillo-Chavez [9] with a constant transfer rate of susceptibles into a partially protected state has been modified to take into account vaccination at birth. The model shows backward bifurcation (existence of multiple endemic stationary states) for certain values of parameters. Parameter values ensuring the existence and nonexistence of endemic equilibria have been discussed. Local and global stability of equilibria have been investigated. The minimum effort required to eradicate the infection has been determined.
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ABSTRACT In this paper, we analyze an SIS epidemic model with partially protective vaccination of efficacy e ∈ [0, 1]. The model exhibits backward bifurcation for certain parameter values. The primary aim of this paper is to investigate... more
ABSTRACT In this paper, we analyze an SIS epidemic model with partially protective vaccination of efficacy e ∈ [0, 1]. The model exhibits backward bifurcation for certain parameter values. The primary aim of this paper is to investigate the possibility of eliminating the infections in static as well as exponentially growing populations with a public health strategy based solely on vaccination. The critical vaccination rate ψ* above which the endemic infection dies out and the conditions on model parameters that ensure its existence are obtained. It has been found that eliminating the infection requires an application of control measures other than vaccination to reduce the basic reproduction number to below the reinfection threshold and then vaccinate susceptible individuals with a rate slightly greater than ψ*. The implication is that, generally, even if all newborns get vaccinated immediately after birth, an effective control is not necessarily assured except if the basic reproduction number is reduced to below the reinfection threshold. We further include the fatality of the infection and investigate its impact on the dynamics. Some numerical simulations are given to illustrate the theoretical analysis.
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Research Interests: Algorithms, Mathematical Biology, Biomathematics, Communicable Diseases, Biological Sciences, and 11 moreHumans, Mathematical Sciences, Animals, Steady state, Survival Rate, Basic Reproduction Number, Reproduction number, Communicable disease control, Epidemic Model, Rate Constant, and Compartmental Model
A discrete worm-load process for a population infected with parasitic infection is considered. It is assumed that initial infections facilitate new infections due to immunosuppression and then there is a limitation due to the immune... more
A discrete worm-load process for a population infected with parasitic infection is considered. It is assumed that initial infections facilitate new infections due to immunosuppression and then there is a limitation due to the immune response. Our model consists of a finite number of partial differential equations which have been used to derive an ordinary differential equation system in the first three cumulants. The deterministic model corresponding to the stochastic one has been derived and analyzed. Conditions on model parameters that ensure the existence of backward bifurcation have been found. The minimum effort required to eliminate the infection has been computed. The impact of neglecting human host’s age structure has been studied and the results showed that ignoring age-structure underestimates the transmission threshold which in turn overestimates the minimum effort required to eliminate the infection. Also, the impact of considering a polygamous mating process of worms wi...
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Research Interests: Epidemiology, Biology, Public Health, Medicine, Temporal dynamics, and 15 moreBiological Sciences, Disease Outbreaks, Humans, Statistical Inference, influenza A, Model Selection, Data Exploration, Renewal Process, Time Factors, Next Generation, Exponential Growth, Growth rate, Reproduction number, Age Structure, and maximum likelihood estimate
Research Interests: Biology and Dissertation
Research Interests: Mathematics and Biology
The two hyper–endemic regions for Visceral Leishmaniasis (VL) in the world are located in India and Sudan. These two countries account for more than half of the world’s VL burden. The regional risk factors associated with VL vary... more
The two hyper–endemic regions for Visceral Leishmaniasis (VL) in the world are located in India and Sudan. These two countries account for more than half of the world’s VL burden. The regional risk factors associated with VL vary drastically per region. A mathematical model of VL transmission dynamics is introduced and parametrized to quantify risk of VL infection in India and Sudan via a careful analysis of VL prevalence level and the control reproductive number,, a metric often used to characterize the degree of endemicity. Parameters, associated with VL-epidemiology for India and Sudan, are estimated using data from health departmental reports, clinical trials, field studies, and surveys in order to assess potential differences between the hyper–endemic regions of India and Sudan. The estimated value of reproduction number for India is found to be 60% higher than that of Sudan (and). It is observed that theis most sensitive to the average biting rate and vector-human transmission...
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A delay differential equations epidemic model of SIQR (SusceptibleInfective-Quarantined-Recovered) type, with arbitrarily distributed periods in the isolation or quarantine class, is proposed. Its essential mathematical features are... more
A delay differential equations epidemic model of SIQR (SusceptibleInfective-Quarantined-Recovered) type, with arbitrarily distributed periods in the isolation or quarantine class, is proposed. Its essential mathematical features are analyzed. In addition, conditions that support the existence of periodic solutions via Hopf bifurcation are identified. Nonexponential waiting times in the quarantine/isolation class lead not only to oscillations but can also support stability switches.
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Research Interests: Epidemiology, Public Health, Temporal dynamics, Biological Sciences, Disease Outbreaks, and 14 moreHumans, Statistical Inference, influenza A, Model Selection, Data Exploration, Renewal Process, Time Factors, Next Generation, Exponential Growth, Growth rate, Reproduction number, Statistical models, Age Structure, and maximum likelihood estimate
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In this paper, we provide a family of ordinary and delay differential equations to describe the dynamics of tumor-growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe... more
In this paper, we provide a family of ordinary and delay differential equations to describe the dynamics of tumor-growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated. The possibility of clearing the tumor, with a strategy, is based on two parameters in the model: the rate of influx of the effector cells, and the rate of influx of IL2. The critical tumor-growth rate, below which endemic tumor does not exist, has been found. One can use the model to make predictions about tumor-dormancy.
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We provide a family of ordinary and delay differential equations to model the dynamics of tumor-growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe under what... more
We provide a family of ordinary and delay differential equations to model the dynamics of tumor-growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated. The possibility of clearing the tumor, with a strategy, is based on two parameters in the model: the rate of influx of the effector cells and the rate of influx of IL-2. The critical tumor-growth rate, below which endemic tumor does not exist, has been found. One can use the model to make predictions about tumor dormancy.
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Background: Regular," moderate", physical exercise is an established non-pharmacological form of treatment for depressive disorders. Brain lateralization has a significant role in the progress of depression. External stimuli... more
Background: Regular," moderate", physical exercise is an established non-pharmacological form of treatment for depressive disorders. Brain lateralization has a significant role in the progress of depression. External stimuli such as various stressors or exercise influence the higher functions of the brain (cognition and affect). These effects often do not follow a linear course. Therefore, nonlinear dynamics seem best suited for modeling many of the phenomena, and putative global pathways in the brain, attributable to such ...
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A delay differential equations epidemic model of SIQR (SusceptibleInfective-Quarantined-Recovered) type, with arbitrarily distributed periods in the isolation or quarantine class, is proposed. Its essential mathematical features are... more
A delay differential equations epidemic model of SIQR (SusceptibleInfective-Quarantined-Recovered) type, with arbitrarily distributed periods in the isolation or quarantine class, is proposed. Its essential mathematical features are analyzed. In addition, conditions that support the existence of periodic solutions via Hopf bifurcation are identified. Nonexponential waiting times in the quarantine/isolation class lead not only to oscillations but can also support stability switches.
Background In many parts of the world, the exponential growth rate of infections during the initial epidemic phase has been used to make statistical inferences on the reproduction number, R, a summary measure of the transmission potential... more
Background In many parts of the world, the exponential growth rate of infections during the initial epidemic phase has been used to make statistical inferences on the reproduction number, R, a summary measure of the transmission potential for the novel influenza A (H1N1) 2009. The growth rate at the initial stage of the epidemic in Japan led to estimates for R in the range 2.0 to 2.6, capturing the intensity of the initial outbreak among school-age children in May 2009. Methods An updated estimate of R that takes into account the epidemic data from 29 May to 14 July is provided. An age-structured renewal process is employed to capture the age-dependent transmission dynamics, jointly estimating the reproduction number, the age-dependent susceptibility and the relative contribution of imported cases to secondary transmission. Pitfalls in estimating epidemic growth rates are identified and used for scrutinizing and re-assessing the results of our earlier estimate of R. Results Maximum likelihood estimates of R using the data from 29 May to 14 July ranged from 1.21 to 1.35. The next-generation matrix, based on our age-structured model, predicts that only 17.5% of the population will experience infection by the end of the first pandemic wave. Our earlier estimate of R did not fully capture the population-wide epidemic in quantifying the next-generation matrix from the estimated growth rate during the initial stage of the pandemic in Japan. Conclusions In order to quantify R from the growth rate of cases, it is essential that the selected model captures the underlying transmission dynamics embedded in the data. Exploring additional epidemiological information will be useful for assessing the temporal dynamics. Although the simple concept of R is more easily grasped by the general public than that of the next-generation matrix, the matrix incorporating detailed information (e.g., age-specificity) is essential for reducing the levels of uncertainty in predictions and for assisting public health policymaking. Model-based prediction and policymaking are best described by sharing fundamental notions of heterogeneous risks of infection and death with non-experts to avoid potential confusion and/or possible misuse of modelling results.
Research Interests: Epidemiology, Public Health, Temporal dynamics, Biological Sciences, Disease Outbreaks, and 13 moreHumans, Statistical Inference, influenza A, Model Selection, Data Exploration, Renewal Process, Time Factors, Next Generation, Exponential Growth, Growth rate, Reproduction number, Age Structure, and maximum likelihood estimate
Research Interests: Epidemiology, Public Health, Temporal dynamics, Biological Sciences, Disease Outbreaks, and 13 moreHumans, Statistical Inference, influenza A, Model Selection, Data Exploration, Renewal Process, Time Factors, Next Generation, Exponential Growth, Growth rate, Reproduction number, Age Structure, and maximum likelihood estimate
The SIS model of Hadeler and Castillo-Chavez [9] with a constant transfer rate of susceptibles into a partially protected state has been modified to take into account vaccination at birth. The model shows backward bifurcation (existence... more
The SIS model of Hadeler and Castillo-Chavez [9] with a constant transfer rate of susceptibles into a partially protected state has been modified to take into account vaccination at birth. The model shows backward bifurcation (existence of multiple endemic stationary states) for certain values of parameters. Parameter values ensuring the existence and nonexistence of endemic equilibria have been discussed. Local and global stability of equilibria have been investigated. The minimum effort required to eradicate the infection has been determined.
Research Interests:
We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model... more
We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R 0 is smaller than the eradication effort for r > 1 + (μ/α) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.