A time-space integro-differential economic model of epidemic control

In this paper we propose a time-space economic model to control the evolution and the spread of a disease. The underlying epidemiological model is formulated as a reaction-diffusion integro-differential partial differential equation. This specific model formulation, supported by empirical data, contains three different terms: a pure diffusion term, a linear growth term, and an integral term. These three terms capture different diffusion channels of a transmissible disease: a local diffusion effect, a temporal effect, and a global diffusion effect. The decision maker aims at deciding the optimal effort to be implemented in order to control the number of infections and, at the same time, minimize the cost of treatment. We analyze the finite horizon case in detail and we provide the closed-form expression of the optimal policy to be implemented to control the epidemic while sustaining economic growth. We also propose two different extensions: The first one considers an infinite horizon model while, the second one, is related to a multi-period framework.