Stability Analysis of an SEIR Model with Treatment

In this paper, an SEIR epidemic model with nonlinear incidence is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number R. If the basic reproduction number \(\mathcal{R}\) < 1, the disease-free equilibrium \(\mathcal{E}\) is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number \(\mathcal{R}\) > 1, the disease is uniformly persistent and the unique endemic equilibrium \(\mathcal{E}\) * of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

In this paper, we establish a discrete-time analog for coupled within-host and between-host systems for an environmentally driven infectious disease with fast and slow two time scales by using the non-standard finite difference scheme. The system is divided into a fast time system and a slow time system by using the idea of limit equations. For the fast system, the positivity and boundedness of the solutions, the basic reproduction number and the existence for infection-free and unique virus infectious equilibria are obtained, and the threshold conditions on the local stability of equilibria are established. In the slow system, except for the positivity and boundedness of the solutions, the existence for disease-free, unique endemic and two endemic equilibria are obtained, and the sufficient conditions on the local stability for disease-free and unique endemic equilibria are established. To return to the coupling system, the local stability for the virus- and disease-free equilibrium, and virus infectious but disease-free equilibrium is established. The numerical examples show that an endemic equilibrium is locally asymptotically stable and the other one is unstable when there are two endemic equilibria.

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).

In this paper, we study a kind of the delayed SEIQR infectious disease model with the quarantine and latent, and get the threshold value which determines the global dynamics and the outcome of the disease. The model has a disease-free equilibrium which is unstable when the basic reproduction number is greater than unity. At the same time, it has a unique endemic equilibrium when the basic reproduction number is greater than unity. According to the mathematical dynamics analysis, we show that disease-free equilibrium and endemic equilibrium are locally asymptotically stable by using Hurwitz criterion and they are globally asymptotically stable by using suitable Lyapunov functions for any Besides, the SEIQR model with nonlinear incidence rate is studied, and the that the basic reproduction number is a unity can be found out. Finally, numerical simulations are performed to illustrate and verify the conclusions that will be useful for us to control the spread of infectious diseases. Meanwhile, the will effect changing trends of in system (1), which is obvious in simulations. Here, we take as an example to explain that.

This paper presents a seven-dimensional ordinary differential equation of mathematical model of zika virus between humans and mosquitoes population with non-linear forces of infection in form of saturated incidence rate. Vertical transmission is introduced into the model. These incidence rates produce antibodies in response to the presence of parasite-causing zika virus in both human and mosquito populations. The existence of region where the model is epidemiologically feasible is established (invariant set) and the positivity of the models is also established. The basic properties of the model are determined including the reproduction number of both cases, R0 and R0 |p=q=0 R respectively. Stability analysis of the disease-free equilibrium is investigated via the threshold parameter (reproduction number R0 |p=q=0) obtained using the next generation matrix technique. The special case model results shown that the disease-free equilibrium is locally asymptotical stable at threshold parameter less than unity and unstable at threshold parameter greater than unity. Under specific conditions on the model parameters, the global dynamics of the special case model around the equilibra are explored using Lyapunov functions. For a threshold parameter less than unity, the disease-free equilibrium is globally asymptotically stable. While the endemic equilibrium is shows to be globally asymptotically stable at threshold parameter greater than unity. Numerical simulations are carried out to confirm the analytic results and explore the possible behavior of the formulated model. The result shows that, horizontal and vertical transmission contributes a higher percentage of infected individuals in the population than only horizontal transmission.

The host migration is one of the important elements that cause the worldwide diffusion and outbreak of many vector-host diseases. In this paper, we formulate a patchy model to investigate the effect of host migration between two patches on the spread of a vector-host disease. The results of the paper show that the reproduction number R0 is a threshold value that determines the uniform persistence and extinction of the disease. If the reproduction number R0 < 1 the disease free equilibrium (DFE) is locally asymptotically stable. If the reproduction number R0 > 1 then the DFE is unstable and the system is uniformly persistent. It is also shown that a unique endemic equilibrium, which exists when R0 > 1, is locally stable if both regions are identical.

The aim of this work is to understand how infectious diseases spread through human populations. Attention is given to those diseases which follow the Susceptible–Infective–Susceptible (SIS) pattern. When modelling diseases spread in a human population, it is important to consider the social and spatial structure of the population. Humans usually live in groups such as work places, households, towns and cities. However, an individual’s membership of a particular group is not fixed. Rather, it changes over time. This structure determines two paths for a disease to spread through the population. Disease is spread between individuals in the same group by contact between infected and susceptible individuals, and is spread from one group to another by the migration of infected individuals. This type of population structure can be modelled by a metapopulation network. I develop a continuous–time Markov chain (CTMC) model that describes the spread of an SIS epidemic in a metapopulation network. I establish an ordinary differential equations (ODE) and a Gaussian diffusion analogue of the stochastic process by applying, respectively, the theory of differential equation approximations for Markov chains, and the theory of density dependent Markov chains. I use the ODE model to derive analytic expressions for various epidemiological quantities of interest. In particular, I obtain expressions for two threshold quantities; the basic reproduction number, and a quantity called T0 which is greater than the basic reproduction number. If the basic reproduction number is above 1, then the disease persists and if the basic reproduction number is below 1, then the disease–free equilibrium (DFE) is locally attractive. However, if T0 is less than or equal to 1, then the DFE is globally attractive. Using the theory of cooperative differential equations and the theory of asymptotically autonomous differential equations, I show the existence and global stability of a unique endemic equilibrium (EE) and the global stability of the DFE in terms of the basic reproduction number, provided that the migration rates of susceptible and infected individuals are equal. Numerical examples indicate that a unique stable EE exists when the condition on the migration rates is relaxed. The approximating Gaussian diffusion shows that the distribution of the population at the endemic level has an approximate multivariate normal distribution whose mean is centered at the endemic equilibrium of the ODE model. The results of this study can serve as a basic framework on how to formulate and analyse a more realistic stochastic model for the spread of an SIS epidemic in a metapopulation which accounts for births, deaths, age, risk, and level of infectivities. Assuming that the model presented here accurately describes the spread of an SIS epidemic in a metapopulation, another question which I address is how to control the spread of the disease. Since most control strategies such as vaccination, treatment and public awareness require a high cost for their implementation, I aim to provide a strategy whose cost is minimal and which only requires control of the migration pattern. Using convex optimisation theory, I obtain an exact analytic expression for the optimal migration pattern for susceptible individuals which minimises the basic reproduction number and the initial growth rate of the epidemic, provided that the migration rate of infected individuals follow a specific pattern. It turns out that the optimal migration pattern for susceptible individuals can be satisfied if the migration rates between any two patches (or groups) are symmetric. The control strategy obtained here can be applied to reduce the early growth rate of a disease in conjunction with or in the absence of another prevention measure.

Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates

In this paper we examine the impact of condom use on the sexual transmission of human immunodeficiency virus (HIV) and acquired immune deficiency syndrome (AIDS) amongst a homogeneously mixing male homosexual population. We first derive a multi-group SIR-type model of HIV/AIDS transmission where the homosexual population is split into subgroups according to frequency of condom use. Both susceptible and infected individuals can transfer between the different groups. We then discuss in detail an important special case of this model which includes two risk groups and perform an equilibrium and stability analysis for this special case. Our analysis shows that this model can exhibit unusual behaviour. As normal, if the basic reproduction number, R0, is greater than unity then there is a unique disease-free equilibrium which is locally unstable and a unique endemic equilibrium. However, when R0 is less than unity two endemic equilibrium solutions can also co-exist simultaneously with the disease-free solution which is locally stable. Numerical simulations using realistic parameter values confirm this and we find that in certain circumstances the disease-free solution and one of the endemic solutions are both locally asymptotically stable, while the other endemic solution is unstable. This unusual behaviour has important implications for control of the disease as reducing R0 to less than unity no longer guarantees eradication of the disease. For a restricted special case of this two-group model we show that there is only the disease-free equilibrium for R0 < or = 1 which is globally stable. For R0 > 1 the disease-free equilibrium is unstable and there is a unique endemic equilibrium which is locally stable. We then attempt to fit the model to HIV and AIDS incidence data from San Francisco, USA. The paper concludes with a brief discussion.

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 &lt; 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 &gt; 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 &gt; 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

Human papillomavirus is a sexually transmitted disease and the major cause of human cancers such as cervical, vulva, rectal, anal, and penile cancer. These cancers lead to the death of many people annually especially cervical cancer in women. This paper presents a deterministic compartmental model for the transmission of human papillomavirus and its progression to cancer. The disease-free equilibrium, reproduction number, and endemic equilibrium are determined. The disease-free equilibrium is both locally and globally asymptotically stable when the vaccination reproduction number (a control parameter) is less than one. The unique endemic equilibrium exists when the vaccination reproduction number exceeds one and the vaccine’s efficacy is negligible. However, the model undergoes a backward bifurcation when the reproduction number approaches one. The bifurcation clears when the vaccine’s efficacy is negligible. With the negligible vaccine efficacy, the endemic equilibrium is locally stable and globally stable. The model is fitted to real cervical cancer data released from Zambia that agreed with the predicted data. The fitted parameters are used for the numerical simulations. The simulation verified the global stability of the disease-free and endemic equilibrium. The numerical simulation results reveal that: different or same-sex contact leads to fewer infections than both contacts, vaccinating girls alone will slightly reduce the susceptibility of adult women but has no effect on the susceptibility of men, vaccinating only girls has an insignificant impact in reducing the number of women with cervical and other HPV-induced cancers, while vaccinating children and adults of both sex show a significant effect in reducing the cervical cancer cases.

Global dynamics of an SEI model with acute and chronic stages

In this paper, an epidemic Susceptible–Vaccinated–Infected–Removed–Susceptible (SVIRS) model is presented on a weighted-undirected network with graph Laplacian diffusion. Disease-free equilibrium always exists while the existence and uniqueness of endemic equilibrium have been shown. When the basic reproduction number is below unity, the disease-free equilibrium is asymptotically globally stable. The endemic equilibrium is asymptotically globally stable if the basic reproduction number is above unity. Numerical analysis is illustrated with a road graph of the state of Minnesota. The effect of all important model parameters has been discussed.

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number [Formula: see text]. If the basic reproduction number [Formula: see text], the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number [Formula: see text], the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

This paper focuses on the study and control of a non linear mathematical epidemic model (S Svih V E LI ) based on a system of ordinary differential equation modeling the spread of tuberculosis infectious with VIH/AIDS coinfection. Existence of both disease free equilibrium and endemic equilibrium are discussed. Reproduction number R0 is determined. Using Lyapunov-Lasalle methods, we analyze the stability of epidemic system around the equilibriums (Disease free and endemic equilibrium). The global asymptotic stability of the disease free equilibrium whenever Rvac < 1 is proved, where R0 is the reproduction number. We prove also that when R0 is less then one, Tuberculosis can be eradicated. Numerical simulations, are conducted to approve analytic results. To achieve control of the disease, seeking to reduce the infectious group by the minimum vaccine coverage we can have. a control problem is formulated. The Pontryagin’s maximum principle is used to characterize the optimal control. The optimality system is derived and solved numericallyusing the Runge Kutta fourth procedure.