Narrow Results

A model for a disease that is transmitted by vectors is formulated. All newborns are assumed susceptible, and human and vector populations are assumed to be constant. The model assumes a saturation effect in the incidences due to the response of the vector to change in the susceptible and infected host densities. Stability of the disease free equilibrium and existence, uniqueness and stability of the endemic equilibrium is investigated. The stability results are given in terms of the basic reproductive number R₀.

A model for the transmission of HIV is proposed with the inclusion of prevention strategies, such as condom use, ARV drug treatment and voluntary counseling and testing. The model is used to predict the potential impact of the strategies that are currently being used to control HIV/AIDS. The model shows that reduction in the number of sexually contacts, by increasing condom use and avoidance of multiple sexual partners has a significant impact in reducing the transmission of the disease.

In this paper, we formulate a vector-borne disease transmission model with a nonlinear incidence and vaccination. The explicit expression of the basic reproduction number R₀(ϕ) which is related to the vaccination rate ϕ is obtained. It has been shown that the global dynamical behavior of the model is completely determined by R₀(ϕ). If R₀(ϕ) < 1, the disease-free equilibrium (DFE) is globally asymptotically stable, and the disease will be eradicated. If R₀(ϕ) > 1, the DFE is unstable, and there exists a unique endemic equilibrium (EE). This equilibrium is globally asymptotically stable which in turn causes the disease to persist in vectors and humans. Finally, a series of numerical simulations, such as sensitive analysis on R₀(ϕ), are performed in order to support the theoretical results.

For some diseases, it is recognized that immunity acquired by natural infection and vaccination subsequently wanes. As such, immunity provides temporal protection to recovered individuals from an infection. An immune period is extended owing to boosting of immunity by asymptomatic re-exposure to an infection. An individual’s immune status plays an important role in the spread of infectious diseases at the population level. We study an age-dependent epidemic model formulated as a nonlinear version of the Aron epidemic model, which incorporates boosting of immunity by a system of delay equations and study the existence of an endemic equilibrium to observe whether boosting of immunity changes the qualitative property of the existence of the equilibrium. We establish a sufficient condition related to the strength of disease transmission from subclinical and clinical infective populations, for the unique existence of an endemic equilibrium.

For a mathematical model for the spread of HIV by sexual transmission in a heterosexual population we analyse the existence and stability of equilibrium solutions. The model is designed to investigate the effects of a fundamental constraint in any social/sexual mixing process for heterogeneous populations. The group contact constraint conserves the number of new sexual partnerships formed per unit time between the sexes, and will have at least a quantitative influence on the dynamics of the model. The analysis has been carried out for general and specific forms of the sexual activity rates (the mean number of sexual partners per unit time for a typical individual). In general we define a threshold parameter R₀, the Reproductive Number, which is a key determinant of the behaviour of the model. We show that in general if R₀>1 there is at least one endemic equilibrium. The specific cases of a dominance sexual activity rate and proportional sexual activity rates are discussed in more detail. Multiple endemic equilibria can occur for the former but not the latter.

The geographic distribution of different viruses has developed widely, giving rise to an escalating number of cases during the past two decades. The deterministic Susceptible, Exposed, Infectious (SEI) models can demonstrate the spatio-temporal dynamics of the diseases and have been used extensively in modern mathematical and mechano-biological simulations. This article presents a functional technique to model the stochastic effects and seasonal forcing in a reliable manner by satisfying the Lipschitz criteria. We have emphasized that the graphical portrayal can prove to be a powerful tool to demonstrate the stability analysis of the deterministic as well as the stochastic modeling. Emphasis is made on the dynamical effects of the force of infection. Such analysis based on the parametric sweep can prove to be helpful in predicting the disease spread in urban as well as rural areas and should be of interest to mathematical biosciences researchers.

A tuberculosis (TB) model with two latent periods, short-term latent period (E₁) and long-term latent period (E₂), and fast and slow progressions is analyzed. The stability of the unique endemic equilibrium of the model is proved. It turns out that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number R₀ ≤ 1, and the endemic equilibrium is globally asymptotically stable if R₀ > 1.

In this paper, an age-structured SEIS epidemic model with infectivity in incubative period is formulated and studied. The explicit expression of the basic reproduction number R₀ is obtained. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀ < 1, at least one endemic equilibrium exists if R₀ > 1. The stability conditions of endemic equilibrium are also given.

An SIS epidemiological model in a population of varying size with two dissimilar groups of susceptible individuals has been analyzed. We prove that all the solutions tend to the equilibria of the system. Then we use the Poincaré Index theorem to determine the number of the rest points and their stability properties. It has been shown that bistability occurs for suitable values of the involved parameters. We use the perturbations of the pitchfork bifurcation points to give examples of all possible dynamics of the system. Some numerical examples of bistability and hysteresis behavior of the system has been also provided.

In this study we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyze a mathematical model that describes the dynamics of HIV infection among the immigrant youths and how parental care can minimize or prevent the spread of the disease in the population. We analyze the model with both screening control and parental care, then investigate its stability and sensitivity behavior. We also conduct both qualitative and quantitative analyses. It is observed that in the absence of infected youths, disease-free equilibrium is achievable and is globally asymptotically stable. We establish optimal strategies for the control of the disease with screening and parental care, and provide numerical simulations to illustrate the analytic results.

Ebola virus disease (EVD) has emerged as a rapidly spreading potentially fatal disease. Several studies have been performed recently to investigate the dynamics of EVD. In this paper, we study the transmission dynamics of EVD by formulating an SEIR-type transmission model that includes isolated individuals as well as dead individuals that are not yet buried. Dynamical systems analysis of the model is performed, and it is consequently shown that the disease-free steady state is globally asymptotically stable when the basic reproduction number, ${{ℛ}}_0$ is less than unity. It is also shown that there exists a unique endemic equilibrium when ${{ℛ}}_0>1$. Using optimal control theory, we propose control strategies, which will help to eliminate the Ebola disease. We use data fitting on models, with and without isolation, to estimate the basic reproductive numbers for the 2014 outbreak of EVD in Liberia and Sierra Leone.

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.