Global dynamics of a time-delayed two-strain epidemic model with general incidence rates and vaccination

This paper investigates a two strains time-delayed epidemic model with general incidence rates and vaccination. Five compartments will be included to the studied model that are the susceptible, the first strain infected subpopulation, the second strain infected ones, the vaccinated and the recovered individuals. Two time delays will be taken into account in our model in order to illustrate the incubation time infection period for each infected strain. In addition, a part of susceptible population is assumed to be vaccinated, but the vaccine is supposed to be efficient against only the first strain. Hence, the vaccinated individuals can be infected only by the second strain only. Since our problem deals with a population dynamics issue, we will demonstrate first that the solution is positive and bounded. In addition to this, the existence result will ensure the wellposedness of our suggested model. The basic reproduction number is given by utilizing the next generation matrix method and five steady states are determined. The global stability of each equilibrium is fulfilled using some suitable Lyapunov functionals. It was demonstrated that the equilibria global stability depends mainly on the first and second strain reproduction numbers. The numerical simulation will end the work by illustrating the equilibria stabilities and highlighting the effect of vaccination. Numerical tests were performed by using come classical biological incidences rates, such as bilinear one, Crowley–Martin, Beddington–DeAngelis and non-monotonic incidence functions. Besides illustrating the equilibria stability, it was shown that the vaccination strategy plays an essential role in controlling the disease spread. The vaccination, even its efficiency against only one strain infected individuals, have demonstrated its importance in eradicating the disease leading to an extinction of the infection. Acting on only one strain infected individuals, by administrating good vaccines, may give good results in terms of stopping the disease spread.