A QUALITATIVE ANALYSIS OF AN A SII R EPIDEMIC MODEL WITH SATURATED INCIDENCE, AWARENESS AND TREATMENT FUNCTIONS

چکیده

An a SII R - type epidemic model with saturated incidence rate affected by inhibitory effect and saturated awareness and treatment functions of infectious individuals is proposed to understand the effect of delays for awareness of disease statuses and treatment of infective individuals on the disease spread. The awareness and treatment functions in this paper are continuous and differential functions which exhibit the effects of delayed awareness and treatment when the rates of awareness and treatment are lower and the number of infected individuals is getting larger. The proposed model has at most two equilibrium points, the disease-free equilibrium (DFE) and endemic equilibrium (EE). The local and global stability conditions of all possible equilibrium points are established. Mathematical analysis reveals that the model has a locally-asymptotically
stable DFE whenever a certain epidemiological threshold, known as the effective reproduction number, , e R is less than unity. However, it is shown that the global asymptotic stability of the DFE was largely dictated by the delays in awareness of disease statuses and treatment of infective individuals; when such delays are weak, then the DFE is globally-asymptotically stable (GAS) when the effective reproductive number is less than unity. When the delays are strong, it is shown that there is the possibility of the existence of the backward bifurcation phenomenon whereby the DFE will co-exist with EE, when the effective reproduction number is less than unity. The backward bifurcation phenomenon existed when the delays occurred either singly or jointly, that is if the delay parameters violate certain threshold conditions. Mathematical analysis provides threshold conditions that allows for the global stability of the EE whenever the associated reproduction number is greater than unity. It was also observed that the model changes its stability behaviour around the endemic equilibrium from stable to unstable as the bifurcation parameter k changes and the system exhibits Hopf