Optimal control strategies for a class of vector borne diseases, exemplified by a toy model for malaria

Abstract

This paper contains a unified review of a set of previous papers by the same authors concerning the mathematical modelling and control of malaria epidemics. The presentation moves from a conceptual mathematical model of malaria transmission in an homogeneous population. Among the key epidemiological features of this model, two-age-classes (child and adult) and asymptomatic carriers have been included. As possible control measures, the extra mortality of mosquitoes due to the use of long-lasting treated mosquito nets (LLINs) and Indoor Residual Spraying (IRS) have been included. By taking advantage of the natural double time scale of the parasite and the human populations, it has been possible to provide interesting threshold results. In particular, key parameters have been identified such that below a threshold level, built on these parameters, the epidemic tends to extinction, while above another threshold level it tends to a nontrivial endemic state. The above model has motivated further analysis when a spatial structure of the relevant populations is added. Inspired by the above, additional model reductions have been introduced, which make the resulting reaction-diffusion system mathematically affordable. Only the dynamics of the infected mosquitoes and of the infected humans has been included, so that a two-component reaction-diffusion system is finally taken. The spread of the disease is controlled by three actions (controls)  implemented in a subdomain of the habitat: killing mosquitoes, treating the infected humans and reducing the contact rate mosquitoes-humans.To start with, the problem of the eradicability of the disease is considered, while the cost of the controls is ignored. We prove that it is possible to decrease exponentially both the human and the vector infective population everywhere in the relevant habitat by acting only in a suitable subdomain. Later the regional control problem of reducing the total cost of the damages produced by the disease, of the controls and of the intervention in a certain subdomain is treated for the finite time horizon case. In order to take the logistic structure of the habitat into account the level set method is used as a key ingredient for describing the subregion of intervention. Here this subregion has been better characterized by both area and perimeter. The authors wish to stress that the target of this paper mainly is to attract the attention of the public health authorities towards an effective and affordable practice of implementation of possible control strategies.