Traveling waves for a diffusive SIR-B epidemic model with multiple transmission pathways

In this work, we consider a diffusive SIR-B epidemic model with multiple transmission pathways and saturating incidence rates. We first present the explicit formula of the basic reproduction number $\mathcal{R}_0$. Then we show that if $\mathcal{R}_0>1$, there exists a constant $c^*>0$ such that the system admits traveling wave solutions connecting the disease-free equilibrium and endemic equilibrium with speed $c$ if and only if $c\geq c^*$. Since the system does not admit the comparison principle, we appeal to the standard Schauder's fixed point theorem to prove the existence of traveling waves. Moreover, a suitable Lyapunov function is constructed to prove the upward convergence of traveling waves