The aim of this paper is to investigate the effectiveness and cost-effectiveness of three malaria preventive measures (use of treated bednets, spray of insecticides and a possible treatment of infective humans that blocks transmission to mosquitoes). For this, we consider a mathematical model for the transmission dynamics of the disease that includes these measures. We first consider the constant control parameters' case, we calculate the basic reproduction number and investigate the existence and stability of equilibria; the model is found to exhibit backward bifurcation. We then assess the relative impact of each of the constant control parameters measures by calculating the sensitivity index of the basic reproductive number to the model's parameters. In the time-dependent constant control case, we use Pontryagin's Maximum Principle to derive necessary conditions for the optimal control of the disease. We also calculate the Infection Averted Ratio (IAR) and the Incremental Cost-Effectiveness Ratio (ICER) to investigate the cost-effectiveness of all possible combinations of the three control measures. One of our findings is that the most cost-effective strategy for malaria control, is the combination of the spray of insecticides and treatment of infective individuals. This strategy requires a 100% effort in both treatment (for 20 days) and spray of insecticides (for 57 days). In practice, this will be extremely difficult, if not impossible to achieve. The second most cost-effective strategy which consists of a 100% use of treated bednets and 87% treatment of infective individuals for 42 and 100 days, respectively, is sustainable and therefore preferable.

Optimal control strategies and cost-effectiveness analysis of a malaria model.

Mathematical modelling of infectious diseases has shown that combinations of isolation, quarantine, vaccine and treatment are often necessary in order to eliminate most infectious diseases. However, if they are not administered at the right time and in the right amount, the disease elimination will remain a difficult task. Optimal control theory has proven to be a successful tool in understanding ways to curtail the spread of infectious diseases by devising the optimal diseases intervention strategies. The method consists of minimizing the cost of infection or the cost of implementing the control, or both. This paper reviews the available literature on mathematical models that use optimal control theory to deduce the optimal strategies aimed at curtailing the spread of an infectious disease.

Optimal control in epidemiology

We derive and analyse a deterministic model for the transmission of malaria disease with mass action form of infection. Firstly, we calculate the basic reproduction number, R(0), and investigate the existence and stability of equilibria. The system is found to exhibit backward bifurcation. The implication of this occurrence is that the classical epidemiological requirement for effective eradication of malaria, R(0)<1, is no longer sufficient, even though necessary. Secondly, by using optimal control theory we derive the conditions under which it is optimal to eradicate the disease and examine the impact of a possible combined vaccination and treatment strategy on the disease transmission. When eradication is impossible, we derive the necessary conditions for optimal control of the disease using Pontryagin's Maximum Principle. The results obtained from the numerical simulations of the model show that a possible vaccination combined with effective treatment regime would reduce the spread of the disease appreciably.

Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity.

In this paper we formulate a mathematical model for malaria–cholera co-infection in order to investigate their synergistic relationship in the presence of treatments. We first analyze the single infection steady states, calculate the basic reproduction number and then investigate the existence and stability of equilibria. We then analyze the co-infection model, which is found to exhibit backward bifurcation. The impact of malaria and its treatment on the dynamics of cholera is further investigated. Secondly, we incorporate time dependent controls, using Pontryagin’s Maximum Principle to derive necessary conditions for the optimal control of the disease. We found that malaria infection may be associated with an increased risk of cholera but however, cholera infection is not associated with an increased risk for malaria. Therefore, to effectively control malaria, the malaria intervention strategies by policy makers must at the same time also include cholera control.

A co-infection model of malaria and cholera diseases with optimal control.

Intelligent Control and Automation

https://www.mapleprimes.com/DocumentFiles/211380_Answer/Optimal_control_and_infectiology.pdf

Optimal control and infectiology: Application to an HIV/AIDS model

https://hal.archives-ouvertes.fr/hal-00819342/document

Exact controllability and complete stabilizability for linear systems in Hilbert spaces

State-dependent impulsive delay-differential equations

Optimal control of discrete distributed systems with delay in state and control: state space theory and HUM approaches

This paper deals with the exact controllability and the quadratic cost control problems for distributed systems with a discrete delayed control. The Hilbert uniqueness method is developed to express the solution of both problems. To illustrate the theory an example of diffusion system is treated.

Jamila Karrakchou

Papers

Optimal control and infectiology: Application to an HIV/AIDS model

Exact controllability and complete stabilizability for linear systems in Hilbert spaces

State-dependent impulsive delay-differential equations

Optimal control of discrete distributed systems with delay in state and control: state space theory and HUM approaches

Exact controllability and optimal control for distributed systems with a discrete delayed control