Stability Of Disease-free Equilibrium Research Articles

TUBERCULOSIS TRANSMISSION MODEL WITH CASE DETECTION AND TREATMENT

A deterministic tuberculosis model for theoretically assessing the potential impact of the combined effects of case detection in the presence of treatment is formulated. The qualitative features of its equilibria are analyzed and it is found that the disease-free equilibrium may not be globally asymptotically stable when the reproduction number is less than unity. This disease threshold number is further used to assess the impact of active TB case finding alone and in conjunction with treatment. A critical threshold parameter say for which case detection will have a positive impact is derived. Using the Centre Manifold theory, the model may exhibit the phenomenon of backward bifurcation (coexistence of a locally stable endemic equilibrium with a stable disease-free equilibrium) when the reproduction number is less than unity. It is shown that the possibility of backward bifurcation occurring decreases with increase case detection. Graphical representations suggest that increase in case finding accompanied by treatment of detected TB cases, result in a marked decrease of TB cases (both latent and active TB).

A Hepatitis B and C Virus Model with Age since Infection that Exhibits Backward Bifurcation

We propose a mathematical model describing the dynamics of hepatitis B or C virus infection. The method of characteristics reduces the age-structured model to a system of differential equations with distributed delay. The model is rigorously analyzed, and a detailed stability analysis is performed. The model, consisting of two mutually exclusive compartments representing the interaction of the virus in the liver and the blood, has a locally asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number $R_0$, is less than unity. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained. Further, under biologically realistic hypotheses, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. This means that reducing the basic reproduction number $R_0$ below 1 is not always sufficient to control HBV/HCV infection when the virus interacts with cells in both the liver and the blood. Numerical analysis confirms the results and stresses the role of some parameters involved in the elimination and persistence of the infection.

A SEASONALLY FORCED EPIDEMIC MODEL WITH TARGETED ANTIVIRAL PROPHYLAXIS

In this paper, we propose a mathematical model to describe the transmission dynamics of infectious diseases with targeted antiviral prophylaxis strategy. Our model incorporates seasonal driving force since seasonal force has a great effect on the spread of infectious diseases. Based on the local stability of disease free equilibrium we derive the control reproduction number [Formula: see text]. Sufficient conditions for the global stability of the disease free equilibrium are obtained. Using the persistence theory for discrete dynamical system, we prove that the infectious disease will remain endemic if [Formula: see text]. Simulation results are also provided to study the effect of targeted antiviral prophylaxis on transmission dynamics of infectious disease and investigate the influence of seasonality on the efficiency of targeted antiviral prophylaxis strategy.

A mathematical model for antimalarial drug resistance

We formulate and analyze a mathematical model for malaria with treatment and the well-known three levels of resistance in humans. The model incorporates both sensitive and resistant strains of the parasites. Analytical results reveal that the model exhibits the phenomenon of backward bifurcation (co-existence of a stable disease-free equilibrium with a stable endemic equilibrium), an epidemiological situation where although necessary, having the basic reproduction number less than unity, it is not sufficient for disease elimination. Through quantitative analysis, we show the effects of varying treatment levels in a high transmission area with different levels of resistance. Increasing treatment has limited benefits in a population with resistant strains, especially in high transmission settings. Thus, in a cost-benefit analysis, the rate of treatment and percentage to be treated become difficult questions to address.

Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate

In this paper, an SEIV epidemic model with vaccination and nonlinear incidence rate is formulated. The analysis of the model is presented in terms of the basic reproduction number R0. It is shown that the model has multiple equilibria and using the center manifold theory, the model exhibits the phenomenon of backward bifurcation where a stable disease-free equilibrium coexists with a stable endemic equilibrium for a certain defined range of R0. We also discuss the global stability of the endemic equilibrium by using a generalization of the Poincare–Bendixson criterion. Numerical simulations are presented to illustrate the results.

Dynamical analysis of a sex-structured Chlamydia trachomatis transmission model with time delay

The effect of using time delay to model the latency period of Chlamydia trachomatis infection is explored, by designing a deterministic two-sex model for Chlamydia transmission dynamics in a population. The resulting delay differential equation model is shown to undergo the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with one or more stable endemic equilibria when the associated reproduction threshold is less than unity. This phenomenon arises due to the re-infection of individuals who recovered from the disease. Using permanence theory, it is shown that Chlamydia will persist in the population whenever the associated reproduction threshold exceeds unity. It is further shown that long latency period could induce positive (decrease disease burden) or negative (increase disease burden) population-level impact depending on the sign of a certain epidemiological threshold quantity and some other conditions. Furthermore, this study shows that adding a time delay (to model the latency period) does not alter the main equilibrium dynamics (with respect to the effective control or persistence of the disease in the community) of the corresponding non-delayed Chlamydia transmission model considered in our earlier study Sharomi and Gumel (2009) [7].

The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay

The problem of the asymptotic dynamics of a quarantine/isolation model with time delay is considered, subject to two incidence functions, namely standard incidence and the Holling type II (saturated) incidence function. Rigorous qualitative analysis of the model shows that it exhibits essentially the same (equilibrium) dynamics regardless of which of the two incidence functions is used. In particular, for each of the two incidence functions, the model has a globally asymptotically stable disease-free equilibrium whenever the associated reproduction threshold quantity is less than unity. Further, it has a unique endemic equilibrium when the threshold quantity exceeds unity. For the case with the Holling type II incidence function, it is shown that the unique endemic equilibrium of the model is globally asymptotically stable for a special case. The permanence of the disease is also established for the model with the Holling type II incidence function. Furthermore, it is shown that adding time delay to and/or replacing the standard incidence function with the Holling type II incidence function in the corresponding autonomous quarantine/isolation model with standard incidence (considered in Safi and Gumel (2010) [10]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease). Finally, numerical simulations of the model with standard incidence show that the disease burden decreases with increasing time delay (incubation period). Furthermore, models with time delay seem to be more suitable for modeling the 2003 SARS outbreaks than those without time delay.

The dynamics of plant disease models with continuous and impulsive cultural control strategies

Plant disease mathematical models including continuous cultural control strategy and impulsive cultural control strategy are proposed and investigated. This novel theoretical framework could result in an objective criterion on how to control plant disease transmission by replanting of healthy plants and removal of infected plants. Firstly, continuous replanting of healthy plants and removing of infected plants is taken. The existence and stability of disease-free equilibrium and positive equilibrium are studied and continuous cultural control strategy is given. Secondly, plant disease model with impulsive replanting of healthy plants and removing of infected plants is also considered. Using Floquet's theorem and small amplitude perturbation, the sufficient conditions under which the infected plant free periodic solution is locally stable are obtained. Moreover, permanence of the system is investigated. Under certain parameter spaces, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations. The modeling methods and analytical analysis presented can serve as an integrating measure to identify and design appropriate plant disease control strategies.

Dynamical behavior of a hepatitis B virus transmission model with vaccination

Hepatitis B virus (HBV) infection is a globally health problem. In 2005, the WHO Western Pacific Regional Office set a goal of reducing chronic HBV infection rate to less than 2% among children five years of age by 2012, as an interim milestone towards the final goal of less than 1%. Many countries made some plans (such as free HBV vaccination program for all neonates in China now) to control the transmission HBV. We develop a model to explore the impact of vaccination and other controlling measures of HBV infection. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R 0 ≤ 1 , and a globally asymptotically stable endemic equilibrium when R 0 > 1 . Numerical simulation results show that the vaccination is a very effective measure to control the infection and they also give some useful comments on controlling the transmission of HBV.

Qualitative assessment of the role of public health education program on HIV transmission dynamics

This paper presents a non-linear deterministic model for assessing the impact of public health education campaign on curtailing the spread of the human immunodeficiency virus (HIV) pandemic in a population. Rigorous qualitative analysis of the model reveals that it exhibits the phenomenon of backward bifurcation (BB), where a stable disease-free equilibrium coexists with a stable endemic equilibrium when a certain threshold quantity, known as the 'effective reproduction number' ('Reff), is less than unity. The epidemiological implication of BB is that a public health education campaign could fail to effectively control HIV even when the classical requirement of having the associated reproduction number less than unity is satisfied. Furthermore, an explicit threshold value is derived above which such an education campaign could lead to detrimental outcome (increase disease burden) and below which it would have positive population-level impact (reduce disease burden in the community). It is shown that the BB phenomenon is caused by imperfect efficacy of the public health education program. The model is used to assess the potential impact of some targeted public health education campaigns using data from numerous countries.

Backward Bifurcation and Optimal Control in Transmission Dynamics of West Nile Virus

The paper considers a deterministic model for the transmission dynamics of West Nile virus (WNV) in the mosquito-bird-human zoonotic cycle. The model, which incorporates density-dependent contact rates between the mosquito population and the hosts (birds and humans), is rigorously analyzed using dynamical systems techniques and theories. These analyses reveal the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity) in WNV transmission dynamics. The epidemiological consequence of backward bifurcation is that the classical requirement of having the reproduction number less than unity, while necessary, is no longer sufficient for WNV elimination from the population. It is further shown that the model with constant contact rates can also exhibit this phenomenon if the WNV-induced mortality in the avian population is high enough. The model is extended to assess the impact of some anti-WNV control measures, by re-formulating the model as an optimal control problem with density-dependent demographic parameters. This entails the use of two control functions, one for mosquito-reduction strategies and the other for personal (human) protection, and redefining the demographic parameters as density-dependent rates. Appropriate optimal control methods are used to characterize the optimal levels of the two controls. Numerical simulations of the optimal control problem, using a set of reasonable parameter values, suggest that mosquito reduction controls should be emphasized ahead of personal protection measures.

Global asymptotic properties of an SEIRS model with multiple infectious stages

The paper presents a rigorous mathematical analysis of a deterministic model, which uses a standard incidence function, for the transmission dynamics of a communicable disease with an arbitrary number of distinct infectious stages. It is shown, using a linear Lyapunov function, that the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium when the threshold exceeds unity. The equilibrium is shown to be locally-asymptotically stable, for a special case, using a Krasnoselskii sub-linearity trick. Finally, a non-linear Lyapunov function is used to show the global asymptotic stability of the endemic equilibrium (for the special case). Numerical simulation results, using parameter values relevant to the transmission dynamics of influenza, are presented to illustrate some of the main theoretical results.

Modelling Tuberculosis and Hepatitis B Co-infections

Tuberculosis (TB) is the leading cause of death among individuals infected with the hepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. We formulate and analyze a deterministic mathematical model which incorporates of the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only and TB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the TB-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for TB, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, that the full HBV-TB co-infection model undergoes a backward bifurcation phenomenon. Through simulations, we mainly find that i) the two diseases will co-exist whenever their partial reproductive numbers exceed unity; (ii) the increased progression rate due to exogenous reinfection from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and (iii) the increased progression rates from acute stage to chronic stage of HBV infection have increased the prevalence levels of HBV and TB prevalences.

Global asymptotic dynamics of a model for quarantine and isolation

The paper presents an SEIQHRS model for evaluating the combined impact of quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms) on the spread of a communicable disease. Rigorous analysis of the model, which takes the form of a deterministic system of nonlinear differential equations with standard incidence, reveal that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. Further, the model has a unique endemic equilibrium when the threshold quantity exceeds unity. Using a Krasnoselskii sub-linearity trick, it is shown that the unique endemic equilibrium is locally-asymptotically stable for a special case. A nonlinear Lyapunov function of Volterra type is used, in conjunction with LaSalle Invariance Principle, to show that the endemic equilibrium is globally-asymptotically stable for a special case. Numerical simulations, using a reasonable set of parameter values (consistent with the SARS outbreaks of 2003), show that the level of transmission by individuals isolated in hospitals play an important role in determining the impact of the two control measures (the use of quarantine and isolation could offer a detrimental population-level impact if isolation-related transmission is high enough).

Qualitative dynamics of a vaccination model for HSV-2

A new mathematical model for the transmission dynamics of herpes simplex virus type 2 (HSV-2), which takes into account disease transmission by infected individuals in the quiescent state and an imperfect HSV-2 vaccine, is designed and qualitatively analysed. In the absence of vaccination, it is shown that the model has a globally asymptotically stable (GAS) disease-free equilibrium (DFE) point whenever an epidemiological threshold, known as the ‘basic reproduction number’, is less than unity. Further, this model has a unique endemic equilibrium whenever the reproduction number exceeds unity. Using a non-linear Lyapunov function, it is shown that the unique endemic equilibrium is GAS (for a special case) when the associated reproduction threshold is greater than unity. On the other hand, the model with vaccination undergoes a vaccine-induced backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium when the reproduction threshold is less than unity. Threshold analysis of the vaccination model reveals that the use of an imperfect HSV-2 vaccine could have positive or negative population-level impact (in reducing disease burden). Simulations of the vaccination model show that an HSV-2 vaccine could lead to effective disease control or elimination if the vaccine efficacy and the fraction of susceptible individuals vaccinated at steady state are high enough (at least 80% each).

Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by R(0)), is less than unity. Further, it has a unique endemic equilibrium if R(0) > 1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by nabla) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by R(vac)) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if R(vac) is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.

Modeling HIV/AIDS and Tuberculosis Coinfection

An HIV/AIDS and TB coinfection model which considers antiretroviral therapy for the AIDS cases and treatment of all forms of TB, i.e., latent and active forms of TB, is presented. We begin by presenting an HIV/AIDS-TB coinfection model and analyze the TB and HIV/AIDS submodels separately without any intervention strategy. The TB-only model is shown to exhibit backward bifurcation when its corresponding reproduction number is less than unity. On the other hand, the HIV/AIDS-only model has a globally asymptotically stable disease-free equilibrium when its corresponding reproduction number is less than unity. We proceed to analyze the full HIV-TB coinfection model and extend the model to incorporate antiretroviral therapy for the AIDS cases and treatment of active and latent forms of TB. The thresholds and equilibria quantities for the models are determined and stabilities analyzed. From the study we conclude that treatment of AIDS cases results in a significant reductions of numbers of individuals progressing to active TB. Further, treatment of latent and active forms of TB results in delayed onset of the AIDS stage of HIV infection.

Re-infection-induced backward bifurcation in the transmission dynamics of Chlamydia trachomatis

A new two-group deterministic model for Chlamydia trachomatis is designed and analyzed to gain insights into its transmission dynamics. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium (DFE) co-exists with one or more stable endemic equilibria when the associated reproduction number is less than unity. It is further shown that the backward bifurcation dynamic is caused by the re-infection of individuals who recovered from the disease. The epidemiological implication of this result is that the classical requirement of the reproduction number being less than unity becomes only a necessary, but not sufficient, condition for disease elimination. The basic model is extended to incorporate the use of treatment for infectious individuals (including those who show disease symptoms and those who do not). Rigorous analysis of the treatment model reveals that the use of treatment could have positive or negative population-level impact, depending on the sign of a certain epidemiological threshold. The treatment model is used to evaluate various treatment strategies, namely treating every infected individual showing symptoms of Chlamydia (universal strategy), treating only infectious males showing Chlamydia symptoms (male-only strategy) and treating only infectious females showing symptoms of Chlamydia (female-only strategy). Numerical simulations show that the implementation of the male-only or female-only strategy can induce an indirect benefit of saving new cases of Chlamydia infection in the opposite sex. Further, the universal strategy gives the highest reduction in the cumulative number of new cases of infection.

Mathematical analysis of a model for HIV-malaria co-infection

A deterministic model for the co-interaction of HIV and malaria in a community is presented and rigorously analyzed. Two sub-models, namely the HIV-only and malaria-only sub-models, are considered first of all. Unlike the HIV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the malaria-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for malaria, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, using centre manifold theory, that the full HIV-malaria co-infection model undergoes backward bifurcation. Simulations of the full HIV-malaria model show that the two diseases co-exist whenever their reproduction numbers exceed unity (with no competitive exclusion occurring). Further, the reduction in sexual activity of individuals with malaria symptoms decreases the number of new cases of HIV and the mixed HIV-malaria infection while increasing the number of malaria cases. Finally, these simulations show that the HIV-induced increase in susceptibility to malaria infection has marginal effect on the new cases of HIV and malaria but increases the number of new cases of the dual HIV-malaria infection.

Global dynamics of a two-strain avian influenza model†

A deterministic model for the transmission dynamics of avian influenza in birds (wild and domestic) and humans is developed. The model, which allows for the transmission of an avian strain and its mutant (assumed to be transmissible between humans), as well as the isolation of individuals with symptoms of any of the two strains, has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. Further, the model has a unique endemic equilibrium whenever this threshold quantity exceeds unity. It is shown, using a non-linear Lyapunov function and LaSalle invariance principle, that this endemic equilibrium is globally asymptotically stable for a special case of the avian-only system. Numerical simulations show that, on average, the isolation of individuals with the avian strain is more beneficial than isolating those with the mutant strain. Furthermore, disease burden increases with increasing mutation rate of the avian strain. In memory of my wonderful father, Alhaji Babandi (‘Zaki’) Gumel (1915–2008)