Mathematical analysis and optimal control of age-structured social interaction model with law enforcement

Optimal use of available resources is one of the major issues in many situations. Apart from this, age is also an important factor in modeling social interaction in a society. In this article, we consider an age-dependent social interaction model and study the stability and optimal control. Law enforcement is an important factor in controlling crime, and the deployment of police to enforce it is important. Now cost is a major issue, so optimal deployment is very important to study. Stability results are derived in terms of threshold parameters. Basic reproduction number is calculated for stability analysis. Using the adjoint system, the form of the law enforcement factor is obtained in terms of state space variables. We see that the cost functional increases with the increase in the population density of criminal population. Numerical results are also added to visually illustrate our theoretical results.