Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

Ardak Kashkynbayev,Fathalla A. Rihan

doi:10.3390/math9151829

Ardak Kashkynbayev, Fathalla A. Rihan

Open Access

https://doi.org/10.3390/math9151829

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Abstract

In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if R0<1, the disease-free steady-state is locally and globally asymptotically stable. However, for R0>1, there exists a positive (endemic) steady-state which is locally and globally asymptotically stable. A Holling type III response function is considered in the numerical simulations to illustrate the effectiveness of the theoretical results.

Highlights

The study of the spread of disease is one of the main directions of mathematical modeling
There has been enormous effort to study the dynamics of infectious disease and various generalizations of the epidemic models that exist in the literature
By the Lyapunov–Lasalle theorem for fractional-order differential equations (FODE) (Lemma 4.6 in [11]), we conclude that E0 (S0, I0 ) is globally asymptotically stable

Summary

Introduction

The study of the spread of disease is one of the main directions of mathematical modeling. The authors in [26] considered a fractionalorder Susceptible-Exposed-Infected-Recovered (SEIR) model with a generalized incidence rate function of the type S f ( I ), where f satisfies some certain conditions. Lahrouz et al studied a fractional-order Susceptible-Infected-Recovered (SIR) model with a general incidence rate function and carried out Mittag–Leffler stability and bifurcation analysis for the model with and without delays [24]. It takes some time for an infected person to be able to transmit a disease to another susceptible host This time-interval is called the latency period, and is modeled by delay differential equations. The study of a so-called Susceptible-Infected-Recovered (SIR) model goes back to the work by Kermack and McKendrick [31] Since this model has been widely studied to predict the dynamics of different infectious diseases. Some preliminaries and definitions about fractional-order derivatives are given in the Appendix A

Fractional Epidemic Model with a General Incidence Rate

Local Stability of the Epidemic Model

Holling Type III Functional Response

Global Stability Analysis

Numerical Simulations

Concluding Remarks