Stability Analysis of a Delayed Fractional Order Sirs Epidemic Model With Nonlinear Incidence Rate

Abstract

In this paper, we study the stability of a fractional order SIRS epidemic model with nonlinear incidence rate and time delay, where the fractional derivative is defined in the Caputo sense. The delay is introduced into the model in order to modeled the incubation period. Using the stability analysis of delayed fractional order systems, we prove that the disease-free equilibrium is locally asymptotically stable when the basic reproduction number R 0 < 1. Also, we show that if R 0 > 1, the endemic equilibrium is locally asymptotically stable.

The impact of dual time delay and Caputo fractional derivative on the long-run behavior of a viral system with the non-cytolytic immune hypothesis

“…That is, fractional-order differential operator is non-local in the sense that it takes into account the fact that the future state not only depends upon the present state but also on the history of its previous states. Obviously, this feature is very relevant for modeling the spread of infections [ [35][36][37][38][39][40][41][42]. For this reason, many researchers have adopted this analytical vision.…”

“…Here, τ 1 indicates the ripeness time of the newly produced viruses and τ 2 designates the required period for achieving the immunity state. Generally, the amalgamation of time delays with fractional derivatives has several advantages and provides excellent insight into the infection dynamics [48][49][50][51]. In accordance with our knowledge, system (3) has not yet been studied due to its complexity.…”

“…It is well known that stability and Hopf bifurcation are two important properties of dynamical systems [52][53][54][55][56]. They give a good overview of the long term of the studied phenomenon [57][58][59]. Unlike classical investigations, the contribution of this survey can be listed as follows:…”

Threshold Dynamics of an $$\mathbf {SEIS}$$ Epidemic Model with Nonlinear Incidence Rates

“…Initially, Capasso and Serio [19] in the study of the cholera epidemic spread in Bari in 1973, observed that the incidence rate may increase more slowly as I increases rather than linearly increasing, and found the saturated incidence rate SI 1+ I which gives good agreement with observations, where the parameter measures the inhibitory effect either due to the crowding caused by high numbers of cases or from the behavioral change of susceptible individuals as their number increases. From then on, a number of authors have studied epidemiological models with several nonlinear incidence rates, see, e.g., [20][21][22][23][24][25][26].…”