Traveling wave solutions in a two-group epidemic model with latent period

In this paper, we propose a susceptible-infective-recovered (SIR) epidemic model to describe the geographic spread of an infectious disease in two groups/sub-populations living in a spatially continuous habitat. It is assumed that the susceptibility of individuals for infection and the infectivity of individuals are distinct between these two groups/sub-populations. It is also assumed that the infectious disease has a fixed latent period and the latent individuals may diffuse. We investigate the traveling wave solutions and obtain complete information about the existence and nonexistence of nontrivial traveling wave solutions. We prove that when the basic reproduction number R-0(S-1(0), S-2(0)) > 1 at the disease free equilibrium (S-1(0), S-2(0),0, 0), there exists a critical number c* > 0 such that for each c > c*, the system admits a nontrivial traveling wave solution with wave speed c, and for c < c*, the system admits no nontrivial traveling wave solution. When R0(S-1(0), S-2(0)) <= 1, we show that there exists no nontrivial traveling wave solution. In addition, for the case R-0(S-1(0), S-2(0)) > 1 and c > c*, we also find that the final sizes of susceptible individuals, denoted by (S-1,S-0, S-2,S-0), satisfies R-0(S-1,S-0, S-2,S-0) < 1, which means that there is no outbreak of this the infectious disease anymore. At last, we analyze and simulate the continuous dependence of the minimal speed c* on the parameters.