Global dynamics of a periodic SEIRS model with general incidence rate

We consider a family of periodic SEIRS epidemic models with a fairly general incidence rate and it is shown the basic reproduction number determines the global dynamics of the models and it is a threshold parameter for persistence. Numerical simulations are per- formed to estimate the basic reproduction number and illustrate our analytical findings, using a nonlinear incidence rate

Dynamics of non-autonomous SEIRS models with general incidence

We consider SEIRS models with general incidence functions depending on the susceptibles, the infectives and the total population, and we analyze this models in several scenarios: autonomous, general non-autonomous and periodic. In all this settings, we discuss the strong persistence and the extinction of the disease. Additionally, we address the following problems: in the autonomous setting, we obtain results on the existence and global stability of disease-free and endemic equilibriums; in the periodic setting, we obtain the global stability of disease-free periodic solution when the basic reproductive number is less than one, and, using the wellknown Mawhin continuation theorem, we discuss the existence of endemic periodic solutions; in the general non-autonomous setting, we prove that our conditions for strong persistence and extinction are robust, in the sense that they are unchanged by su ciently small perturbations of the parameters and the incidence functions. Finally, we consider a version of our model with two control variables, vaccination and treatment, and study the existence and uniqueness of solution of the optimal control model considered. Some computational experiences illustrate our results.Consideramos modelos SEIRS com funções de incidência gerais dependendo dos suscetíveis, dos infeciosos e da população total, e analisamos esses modelos em diversos cenários: autónomo, não-autónomo geral e periódico. Em todas essas situações, analisamos a persistência forte e a extinção da doença. Além disso, abordamos os seguintes problemas: no caso autónomo, obtemos resultados sobre a existência e a estabilidade global do equilíbrio livre de doença e do equilíbrio endémico; no caso periódico, obtemos a estabilidade global da solução periódica livre de doença quando o número reprodutivo básico é inferior a um, e, usando o conhecido teorema de continua ção de Mawhin, discutimos a existência de soluções periódicas endémicas; no caso não-autónomo geral, provamos que as nossas condições para persistência forte e extinção são robustas, no sentido em que se mantêm inalteradas para perturbações su cientemente pequenas dos parâmetros e das funções de incidência. Finalmente, consideramos uma versão do nosso modelo com duas variáveis de controle, vacinação e tratamento, e estudamos a existência e unicidade da solução do modelo de controle ótimo considerado. Algumas experiências computacionais ilustram os nossos resultados

Nonlinear incidence and stability of infectious disease models

In this paper we consider the impact of the form of the non-linearity of the infectious disease incidence rate on the dynamics of epidemiological models. We consider a very general form of the non-linear incidence rate (in fact, we assumed that the incidence rate is given by an arbitrary function f (S, I, N) constrained by a few biologically feasible conditions) and a variety of epidemiological models. We show that under the constant population size assumption, these models exhibit asymptotically stable steady states. Precisely, we demonstrate that the concavity of the incidence rate with respect to the number of infective individuals is a sufficient condition for stability. If the incidence rate is concave in the number of the infectives, the models we consider have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case the infection-free equilibrium state is stable. For the incidence rate of the form g(I)h(S), we prove global stability, constructing a Lyapunov function and using the direct Lyapunov method. It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals. We demonstrate this result using a SIRS model and a SEIRS model as case studies. For other compartment epidemic models, the analysis is quite similar, and the same conclusion, namely stability of the equilibrium states, holds

Mathematical Modeling, Simulation, and Time Series Analysis of Seasonal Epidemics.

Seasonal and non-seasonal Susceptible-Exposed-Infective-Recovered-Susceptible (SEIRS) models are formulated and analyzed. It is proved that the disease-free steady state of the non-seasonal model is locally asymptotically stable if Rv \u3c 1, and disease invades if Rv \u3e 1. For the seasonal SEIRS model, it is shown that the disease-free periodic solution is locally asymptotically stable when R̅v \u3c 1, and I(t) is persistent with sustained oscillations when R̅v \u3e 1. Numerical simulations indicate that the orbit representing I(t) decays when R̅v \u3c 1 \u3c Rv. The seasonal SEIRS model with routine and pulse vaccination is simulated, and results depict an unsustained decrease in the maximum of prevalence of infectives upon the introduction of routine vaccination and a sustained decrease as pulse vaccination is introduced in the population. Mortality data of pneumonia and influenza is collected and analyzed. A decomposition of the data is analyzed, trend and seasonality effects ascertained, and a forecasting strategy proposed

Existence of periodic solutions of a periodic SEIRS model with general incidence.

For a family of periodic SEIRS models with general incidence, we prove the existence of at least one endemic periodic orbit when some condition related to R0 holds. Additionally, we prove the existence of a unique disease-free periodic orbit, that is globally asymptotically stable when R0 < 1. In particular, our main result generalizes the one in Zhang et al. (2012). We also discuss some examples where our results apply and show that, in some particular situations, we have a sharp threshold between existence and non existence of an endemic periodic orbit

A survey on Lyapunov functions for epidemic compartmental models

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey–predator or rumor spreading

Global Stability of Worms in Computer Network

An attempt has been made to show the impact of non-linearity of the worms through SIRS (susceptible – infectious – recovered - susceptible) and SEIRS (susceptible – exposed – infectious – recovered - susceptible) e-epidemic models in computer network. A very general form of non-linear incidence rate has been considered satisfying the worm propagating behavior in computer network. The concavity conditions with a non-linear incidence rate and under the constant population size assumption are shown to be stable. Such systems have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case, the worm infection-free equilibrium is stable

Optimal control of non-autonomous SEIRS models with vaccination and treatment

We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled modelspublishe