Epidemic Model with Direct and Indirect Transmission modes and Two Delays

We propose an epidemic model in which all diseases are transmitted by direct transmission (infectives to susceptibles) and indirect transmission which occurs through shedding of virus by infectives and acquisition by susceptibles. The model takes into account the effect of latency period and time needed for a susceptible to become infective by indirect contact. Under the certain assumptions, the basic reproduction number of the model is identified from the direct and indirect basic reproduction numbers. The main goal is to analyse the asymptotic behaviours, global stability, bifurcation and to detecte the most sensitive parameters. Numerical simulations are carried out to illustrate the theoretical part.