Global stability of a fractional-order logistic growth model with infectious disease

Abstract

Infectious disease has an influence on the density of a population. In this paper, a fractional-order logistic growth model with infectious disease is formulated. The population grows logistically and divided into two compartments i.e. susceptible and infected populations. We start by investigating the existence, uniqueness, non-negativity, and boundedness of solutions. Furthermore, we show that the model has three equilibrium points namely the population extinction point, the disease-free point, and the endemic point. The population extinction point is always a saddle point while others are conditionally asymptotically stable. For the non-trivial equilibrium points, we successfully show that the local and global asymptotic stability have the similar properties. Especially, when the endemic point exists, it is always globally asymptotically stable. We also show the existence of forward bifurcation in our model. We portray some numerical simulations consist of the phase portraits, time series, and a bifurcation diagram to validate the analytical findings.

The Dynamics of a Discrete Fractional-Order Logistic Growth Model with Infectious Disease

“…These fixed points are similar with its fractional-order model ( 1) in [6]. Now, we study the dynamics of these fixed points by considering the influence of the given step size (ℎ) occurs from the PWCA process.…”

“…The fractional-order logistic growth model with infectious disease proposed by Panigoro & Rahmi [6] is given by the following equations.…”

“…) which is exists if 0 > 1 where 0 is the ratio reproduction number given by 0 = + , see [6].…”

“…The PWCA scheme can be found in [15] and [16]. In this paper, we investigate the dynamics of a discrete model constructed by PWCA from the fractional-order model proposed by Panigoro & Rahmi [6]. As far as we know, the discrete form of this model has never been studied and investigated.…”

“…The different lies on the growth rate of susceptible population and the ability to recover of infected population. Model in [12] using constant growth rate and the infected population cannot recover from the disease while model in [6] using logistic growth rate and the infect population has the ability to recover.…”