Impulsive control and global stabilization of reaction‐diffusion epidemic model

In this paper, by using the variational method, a sufficient condition for the unique existence of the stationary solution of the reaction‐diffusion ecosystem is obtained, which directly leads to the global asymptotic stability of the unique equilibrium point. Moreover, delayed feedback ecosystem with reaction‐diffusion item is considered, and the above sufficient condition may not warrant the global stability for it. Utilizing impulse control results in its globally exponential stability. Besides, one of numerical examples shows that applying Laplacian semigroup in a newly obtained criterion is better than applying impulsive differential inequations, for the impulse condition becomes less harsh. It is worth mentioning that the Neumann zero‐boundary value that the infected and the susceptible people or animals should be controlled in the epidemic prevention area and not allowed to cross the border, which is a good simulation of the actual situation of epidemic prevention. And numerical simulations indicate the effectiveness of impulse control, which has a certain enlightening effect on the actual epidemic prevention work. That is, in the face of the epidemic situation, taking a certain frequency of positive and effective epidemic prevention measures is conducive to the stability and control of the epidemic situation. Particularly, the newly obtained theorems quantifies this feasible step.