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 malariaВ transmission in an homogeneous population. Among the key epidemiological features of thisВ model, two-age-classes (child and adult) and asymptomaticВ carriers have been included. As possible control measures, the extra mortality of mosquitoes due to the use of long-lasting treated mosquito nets (LLINs) andВ Indoor Residual Spraying (IRS) have been included.В By taking advantage of the natural doubleВ time scale of the parasite and the human populations, it hasВ been possible to provide interesting threshold results.В 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 analysisВ 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.В Only the dynamics of theВ infected mosquitoes and of the infected humans has been included, so that a two-component reaction-diffusion system is finally taken.В The spread of theВ disease is controlled by three actions (controls)В  implemented in aВ subdomain of the habitat: killing mosquitoes, treating theВ 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 proveВ that it is possible to decrease exponentially both the humanВ and the vector infective population everywhere in the relevantВ habitat by acting only in a suitable subdomain. Later theВ regional control problem of reducing the total cost of theВ damages produced by the disease, of the controls and of theВ intervention in a certain subdomain is treated for the finite timeВ horizon case. In order to take theВ logistic structure of the habitat into account theВ level set method is used as a key ingredient for describing theВ subregion of intervention. Here this subregion has been better characterized by both area and perimeter.В The authors wish to stress that the target of this paper mainly is to attract the attention of the public health authorities towards anВ effective and affordable practice of implementation of possible control strategies.