Untangling the memory and inhibitory effects on SIS-epidemic model with Beddington–DeAngelis infection rate

Abstract

The dynamical behaviors of an epidemic model based on the susceptible–infected–susceptible (SIS) model are investigated. The Beddington–DeAngelis functional response is used for the infection rate to present the dependence of the transmission of the infection on the ratio of both susceptible and infected populations. A Caputo fractional derivative is applied to show the existence of memory in nature affects population dynamics. The disease-free and endemic equilibrium points are obtained as the equilibrium points that describe the condition when the population is free from the disease or the disease exists in the population throughout time. The existence, uniqueness, non-negativity, and boundedness are proven which state the biological validity of the mathematical model. The local and global stability of each equilibrium point is studied including the basic reproduction number and its influence on the dynamical behaviors. Some numerical simulations are portrayed to explore more about the dynamics of the model which are relevant to the analytical findings. The partial rank correlation coefficient is presented to investigate the dominant parameter with respect to the basic reproduction number and the density of susceptible and infected populations. The parameter continuations are demonstrated to show the impact of infection rate and the inhibitory effect which lead to the occurrence of forward bifurcations. The memory effect is also demonstrated numerically to show the changes in convergence rate due to the changes in memory strength.