Dynamical properties of a new SIR epidemic model

Taking into account the depletion of food supply by all individuals, and the fact that chronic infectious diseases will not cause the infected individuals to lose their fertility completely, we first propose a new SIR epidemic model of ODE. For this model, we derive its basic reproduction number $ \mathcal{R}_0 $, and show that the disease-free equilibrium point is globally asymptotically stable when $ \mathcal{R}_0\le1 $, while the unique positive equilibrium point is globally asymptotically stable when $ \mathcal{R}_0>1 $. Then we incorporate the spatial dispersion and free boundary condition into this ODE model. The well-posedness and longtime behaviors are obtained. Particularly, we find a spreading-vanishing dichotomy in which the basic reproduction number $ \mathcal{R}_0 $ plays a crucial role.