Optimal vaccination in a SIRS epidemic model Optimal vaccination in a SIRS epidemic model

• Published in: Economic theory Vol. 77; no. 1; pp. 49 - 74
• Main Authors: Federico, Salvatore, Ferrari, Giorgio, Torrente, Maria-Laura
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2024; Springer; Springer Nature B.V
• Subjects: Case studies; Communicable diseases; Economic theory; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Economics and Finance; Epidemics; Game Theory; Immunization; Infectious diseases; Microeconomics; Public Finance; Research Article; Social and Behav. Sciences; Vaccination; Verification; Epidemic; SIRS model; I12; I18; 92D30; Optimal control; Viscosity solution; 93C15; Optimal vaccination; 49K15; 49L25; Non-smooth verification theorem; C61
• ISSN: 0938-2259; 1432-0479; 1432-0479
• DOI: 10.1007/s00199-022-01475-9

We propose and solve an optimal vaccination problem within a deterministic compartmental model of SIRS type: the immunized population can become susceptible again, e.g. because of a not complete immunization power of the vaccine. A social planner thus aims at reducing the number of susceptible individuals via a vaccination campaign, while minimizing the social and economic costs related to the infectious disease. As a theoretical contribution, we provide a technical non-smooth verification theorem, guaranteeing that a semiconcave viscosity solution to the Hamilton–Jacobi–Bellman equation identifies with the minimal cost function, provided that the closed-loop equation admits a solution. Conditions under which the closed-loop equation is well-posed are then derived by borrowing results from the theory of Regular Lagrangian Flows . From the applied point of view, we provide a numerical implementation of the model in a case study with quadratic instantaneous costs. Amongst other conclusions, we observe that in the long-run the optimal vaccination policy is able to keep the percentage of infected to zero, at least when the natural reproduction number and the reinfection rate are small.