Global Analysis of Two Tuberculosis Models

Two models for tuberculosis (TB) that include treatment of latent and infective individuals are considered. The first model assumes constant recruitment with a fixed fraction entering each class, having the consequences that TB never dies out and that the usual threshold condition does not apply. The unique endemic equilibrium is locally asymptotically stable for all parameter values and is shown to be globally asymptotically stable under certain parameter restrictions. The second model has a general recruitment function, but all recruitment is into the susceptible class. Three threshold parameters determine the existence and local stability of equilibria. The disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than or equal to one. The endemic equilibrium, when it exists, is shown to be globally asymptotically stable under certain parameter restrictions. Global stability results for the endemic equilibria are proved using the geometric approach of Li and Muldowney.     

Global stability of a delayed SEIRS epidemic model with saturation incidence rate

In this paper, an SEIRS epidemic model with a saturation incidence rate and a time delay describing a latent period is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is established. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Numerical simulations are carried out to illustrate the main theoretical results.     

Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea

Prior studies have indicated that heavy alcohol drinkers are likely to engage in risky sexual behaviours and thus, more likely to get sexually transmitted infections (STIs) than social drinkers. Here, we formulate a deterministic model for evaluating the impact of heavy alcohol drinking on the reemerging gonorrhea epidemic. The model is rigorously analysed, showing the existence of a globally asymptotically stable disease-free equilibrium whenever the reproductive number is less than unity. If the disease threshold number is greater than unity, a unique endemic equilibrium exists and is globally asymptotically stable in the interior of the feasible region and the disease persists at endemic proportions if it is initially present. Both analytical and numerical results are provided to ascertain whether heavy alcohol drinking has an impact on the transmission dynamics of gonorrhea.     

On the Role of Variable Latent Periods in Mathematical Models for Tuberculosis

The qualitative behaviors of a system of ordinary differential equations and a system of differential-integral equations, which model the dynamics of disease transmission for tuberculosis (TB), have been studied. It has been shown that the dynamics of both models are governed by a reproductive number. All solutions converge to the origin (the disease-free equilibrium) when this reproductive number is less than or equal to the critical value one. The disease-free equilibrium is unstable and there exists a unique positive (endemic) equilibrium if the reproductive number exceeds one. Moreover, the positive equilibrium is stable. Our results show that the qualitative behaviors predicted by the model with arbitrarily distributed latent stage are similar to those given by the TB model with an exponentially distributed period of latency.     

Periodic solutions of an epidemic model with saturated treatment

Based on the fact that many infectious diseases exhibit periodic fluctuations and there is a saturated phenomenon during disease treatment, we study an SIR model with periodic incidence rate and saturated treatment function. Firstly, we find that the basic reproduction number less than 1 cannot insure the global stability of disease-free equilibrium and it needs to add other conditions. Moreover, we establish sufficient conditions for the multiplicity of positive periodic solutions. We also apply the numerical method to confirm theoretical results and show the stability of the periodic solutions. We observe that there are two periodic solutions in the system where one is stable and the other one is unstable. These results will provide some guidance for control measures of disease.     

Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality

A deterministic model of tuberculosis without and with seasonality is designed and analyzed into its transmission dynamics. We first present and analyze a tuberculosis model without seasonality, which incorporates the essential biological and epidemiological features of the disease. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated basic reproduction number is less than unity. The statistical data of tuberculosis (TB) cases show seasonal fluctuations in many countries. Then, the extension of our TB model by incorporating seasonality is developed and the basic reproduction ratio is defined. Parameter values of the model are estimated according to demographic and epidemiological data in Cameroon. The simulation results are in good accordance with the seasonal variation of the reported cases of active TB in Cameroon.     

The importance of immune responses in a model of hepatitis B virus

In this paper, the dynamical behavior of a hepatitis B virus model with CTL immune responses is studied. Analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable if the basic reproductive ratio of virus is less than one and the endemic equilibrium is locally asymptotically stable if the basic reproductive ratio is greater than one. When the basic reproductive ratio is greater than one, the system is uniformly persistent, which means the virus is endemic. Mathematical analysis and numerical simulations show that the CTL immune responses play a significant and decisive role in eradication of disease. The study and information derived from this model may have an important impact on treatment protocols of hepatitis B virus in the future.     

Threshold dynamics for a HFMD epidemic model with?periodic transmission rate

In this paper, a periodic epidemic model is proposed in order to simulate the dynamics of HFMD transmission. We consider the effects of quarantine in the children population. We obtain a threshold value which determines the extinction and uniform persistence of the disease. Our results show that the disease-free equilibrium is globally asymptotically stable if the threshold value is less than unity. Otherwise, the system has a positive periodic solution and the disease persists. Numerical simulations show that quarantine has a positive impact on the spread of disease, i.e., quarantine is beneficial to the intervention and control of the disease outbreak in the children population.     

Assessing the potential impact of immunity waning on the dynamics of COVID-19 in South Africa: an endemic model of COVID-19

We developed an endemic model of COVID-19 to assess the impact of vaccination and immunity waning on the dynamics of the disease. Our model exhibits the phenomenon of backward bifurcation and bi-stability, where a stable disease-free equilibrium coexists with a stable endemic equilibrium. The epidemiological implication of this is that the control reproduction number being less than unity is no longer sufficient to guarantee disease eradication. We showed that this phenomenon could be eliminated by either increasing the vaccine efficacy or by reducing the disease transmission rate (adhering to non-pharmaceutical interventions). Furthermore, we numerically investigated the impacts of vaccination and waning of both vaccine-induced immunity and post-recovery immunity on the disease dynamics. Our simulation results show that the waning of vaccine-induced immunity has more effect on the disease dynamics relative to post-recovery immunity waning and suggests that more emphasis should be on reducing the waning of vaccine-induced immunity to eradicate COVID-19.

     

Global properties of a class of virus infection models with multitarget cells

In this paper, we propose a class of virus infection models with multitarget cells and study their global properties. We first study three models with specific forms of incidence rate function, then study a model with a more general nonlinear incidence rate. The basic model is a (2n+1)-dimensional nonlinear ODEs that describes the population dynamics of the virus, n classes of uninfected target cells, and n classes of infected target cells. Model with exposed state and model with saturated infection rate are also studied. For these models, Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of these models. We have proven that if the basic reproduction number is less than unity then the uninfected steady state is globally asymptotically stable, and if the basic reproduction number is greater than unity then the infected steady state is globally asymptotically stable. For the model with general nonlinear incidence rate, we construct suitable Lyapunov functions and establish the sufficient conditions for the global stability of the uninfected and infected steady states of this model.     

Global analysis of some epidemic models with general contact rate and constant immigration

An epidemic models of SIR type and SIRS type with general contact rate and constant immigration of each class were discussed by means of theory of limit system and suitable Liapunov functions. In the absence of input of infectious individuals, the threshold of existence of endemic equilibrium is found. For the disease-free equilibrium and the endemic equilibrium of corresponding SIR model, the sufficient and necessary conditions of global asymptotical stabilities are all obtained. For corresponding SIRS model, the sufficient conditions of global asymptotical stabilities of the disease-free equilibrium and the endemic equilibrium are obtained. In the existence of input of infectious individuals, the models have no disease-free equilibrium. For corresponding SIR model, the endemic equilibrium is globally asymptotically stable ; for corresponding SIRS model, the sufficient conditions of global asymptotical stability of the endemic equilibrium are obtained.     

Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments

The basic reproduction ratio and its computation formulae are established for a large class of compartmental epidemic models in periodic environments. It is proved that a disease cannot invade the disease-free state if the ratio is less than unity and can invade if it is greater than unity. It is also shown that the basic reproduction number of the time-averaged autonomous system is applicable in the case where both the matrix of new infection rate and the matrix of transition and dissipation within infectious compartments are diagonal, but it may underestimate and overestimate infection risks in other cases. The global dynamics of a periodic epidemic model with patch structure is analyzed in order to study the impact of periodic contacts or periodic migrations on the disease transmission.      

Domains of attraction for multiple limit cycles of coupled van der pol equations by simple cell mapping

One of the most difficult tasks in non-linear analysis is to determine globally various domains of attraction in the state space when there exist more than one asymptotically stable equilibrium states and/or periodic motions. The task is even more demanding if the order of the system is higher than two. In this paper we consider two coupled van der Pol oscillators which admit two asymptotically stable limit cycles. For systems of this kind we show how the method of cell-to-cell mapping can be used to determine the two four-dimensional domains of attraction of the two stable limit cycles in a very effective way. The final results are shown in this paper in the form of a series of graphs which are various two-dimensional sections of the four-dimensional state space.     

The pulse treatment of computer viruses: a modeling study

Unlike new medical procedures, new antivirus software can be disseminated rapidly through the Internet and takes effect immediately after it is run. As a result, a considerable number of infected computers can be cured almost simultaneously. Consequently, it is of practical importance to understand how pulse treatment affects the spread of computer viruses. For this purpose, an impulsive malware propagation model is proposed. To the best of our knowledge, this is the first computer virus model that takes into account the effect of pulse treatment. The dynamic properties of this model are investigated comprehensively. Specifically, it is found that (a) the virus-free periodic solution is globally asymptotically stable when the basic reproduction ratio (BRR) is less than unity, (b) infections are permanent when the BRR exceeds unity, and (c) a locally asymptotically stable viral periodic solution bifurcates from the virus-free periodic solution when the BRR goes through unity. A close inspection of the influence of different model parameters on the BRR allows us to suggest some feasible measures of eradicating electronic infections.     

Global asymptotic stability for Hopfield-type neural networks with diffusion effects

The existence, uniqueness and global asymptotic stability for the equilibrium of Hopfield-type neural networks with diffusion effects are studied. When the activation functions are monotonously nondecreasing, differentiable, and the interconnected matrix is related to the Lyapunov diagonal stable matrix, the sufficient conditions guaranteeing the existence of the equilibrium of the system are obtained by applying the topological degree theory. By means of constructing the suitable average Lyapunov functions, the global asymptotic stability of the equilibrium of the system is also investigated. It is shown that the equilibrium (if it exists) is globally asymptotically stable and this implies that the equilibrium of the system is unique.     

Misfit dislocation dipoles in wire composite solids

Developed in this note is a theoretical model describing the mobility of a misfit screw dislocation dipole in a wire composite consisting of a stiff cylindrical substrate covered by a soft co-axial cylindrical film. A critical value of the film thickness, which is a function of the parameter measuring the stiffness of the film with respect to the substrate, is identified. It is observed that: (i) there exist two equilibrium positions of the misfit dislocation dipole (one stable and the other one unstable) when the film is thicker than the critical value; (ii) the two equilibrium positions of the misfit dislocation dipole converge to a single saddle point equilibrium position which is neither stable nor unstable when the thickness of the film is at the critical value; (iii) there exists no equilibrium position of the misfit dislocation dipole when the thickness of the film is below the critical value. These features could be useful to the design of wire composites and to the dislocation-related plasticity analysis.     

The Impact of Media on the Control of Infectious Diseases

We develop a three dimensional compartmental model to investigate the impact of media coverage to the spread and control of infectious diseases (such as SARS) in a given region/area. Stability analysis of the model shows that the disease-free equilibrium is globally-asymptotically stable if a certain threshold quantity, the basic reproduction number (), is less than unity. On the other hand, if , it is shown that a unique endemic equilibrium appears and a Hopf bifurcation can occur which causes oscillatory phenomena. The model may have up to three positive equilibria. Numerical simulations suggest that when and the media impact is stronger enough, the model exhibits multiple positive equilibria which poses challenge to the prediction and control of the outbreaks of infectious diseases. Research supported by the NNSF of China (10471066). Research supported by NSERC, MITACS and CFI/OIT of Canada.     

Global stability of a virus dynamics model with Beddington–DeAngelis incidence rate and CTL immune response

In this paper, the global stability of virus dynamics model with Beddington–DeAngelis infection rate and CTL immune response is studied by constructing Lyapunov functions. We derive the basic reproduction number R ₀ and the immune response reproduction number R ⁰ for the virus infection model, and establish that the global dynamics are completely determined by the values of R ₀. We obtain the global stabilities of the disease-free equilibrium E ₀, immune-free equilibrium E ₁ and endemic equilibrium E ^∗ when R ₀≤1, R ₀>1, R ⁰>1, respectively.     

Control problems of a mathematical model for schistosomiasis transmission dynamics

Drug treatment, snail control, cercariae control, improved sanitation and health education are the effective strategies which are used to control the schistosomiasis. In this paper, we formulate a deterministic model for schistosomiasis transmission dynamics in order to explore the role of the several control strategies. The basic reproductive number is computed. Sufficient conditions for the global asymptotic stability of the disease-free equilibrium are obtained. By using the Center Manifold Theory, we analyze the local stability of endemic equilibrium. Finally, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproductive number to the changes of control parameters are shown. Our results imply that snail-killing is the most effective way to control the transmission of schistosomiasis.