QUALITATIVE ANALYSIS OF AN SEIS EPIDEMIC MODEL WITH NONLINEAR INCIDENCE RATE

By means of limit theory and Fonda's theorem, an SEIS epidemic model with constant recruitment and the disease-related rate is considered. The incidence term is of the nonlinear form, and the basic reproduction number is found. If the basic reproduction number is less than one, there exists only the disease-free equilibrium, which is globally asymptotically stable, and the disease dies out eventually. If the basic reproduction number is greater than one, besides the unstable disease-free equilibrium, there exists also a unique endemic equilibrium, which is locally asymptotically stable, and the disease is uniformly persistent.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Asymptotic Behavior of the Principal Eigenvalue for Cooperative Elliptic Systems and Applications

The asymptotic behavior of the principal eigenvalue for general linear cooperative elliptic systems with small diffusion rates is determined. As an application, we show that if a cooperative system of ordinary differential equations has a unique positive equilibrium which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann boundary condition or the Robin boundary condition also has a unique positive steady state which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. Moreover, as the diffusion coefficients approach zero, the positive steady state of the reaction-diffusion system converges uniformly to the equilibrium of the corresponding kinetic system.     

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.     

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.     

Propagation of computer virus under the influences of infected external computers and removable storage media

In reality, a portion of infected external computers could enter the Internet, and removable storage media could carry virus. To our knowledge, nearly all previous models describing the spread of computer virus ignore the combined impact of these two factors. In this paper, a new dynamical model is established based on these facts. A systematic analysis of the model is performed, and it is found that the unique (viral) equilibrium is globally asymptotically stable. Some simulation experiments are also made to justify the model. Finally, a result and some applicable measures for suppressing viral spread are suggested.     

Delayed nonlinear cournot and bertrand dynamics with product differentiation

Dynamic duopolies will be examined with product differentiation and isoelastic price functions. We will first prove that under realistic conditions the equilibrium is always locally asymptotically stable. The stability can however be lost if the firms use delayed information in forming their best responses. Stability conditions are derived in special cases, and simulation results illustrate the complexity of the dynamism of the systems. Both price and quantity adjusting models are discussed.     

The spread of computer virus under the effect of external computers

In reality, the external computers, in particular, external infected computers are connected to the Internet. Based on this reasonable assumption, a new computer virus propagation model is established. Different from all the previous models, this model regards the external computers as a single compartment to study. Through a qualitative analysis, it is found that (1) this model possesses a unique (viral) equilibrium, and (2) this equilibrium is globally asymptotically stable. Further study shows that, by taking effective measures, the number of infected computers can be made below an acceptable threshold.     

Global stability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response

A Holling type predator-prey model with stage structure for the predator and a time delay due to the gestation of the mature predator is investigated. By analyzing the characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the model is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium is feasible. By using Lyapunov functionals and the LaSalle invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

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.

     

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.