Deterministic and Stochastic Optimal Control

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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.

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

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

In this study we consider a mathematical model of an SIR epidemic model with a saturated incidence rate. We used the optimal vaccination strategies to minimize the susceptible and infected individuals and to maximize the number of recovered individuals. We work in the nonlinear optimal control framework. The existence result was discussed. A characterization of the optimal control via adjoint variables was established. We obtained an optimality system that we sought to solve numerically by a competitive Gauss–Seidel like implicit difference method.

/pdf/optimal-control-of-an-epidemic-model-with-a-saturated-1q02aaf73y.pdf

Optimal control of an epidemic model with a saturated incidence rate

In this paper, an SIR spatiotemporal epidemic model is formulated as a system of parabolic partial differential equations with no-flux boundary conditions. Immunity is forced through vaccine distribution considered a control variable. Our principal objective is to characterize an optimal control that minimizes the number of infected individuals and the costs associated with vaccination over a finite space and time domain. The existence of solutions to the state system and the existence of an optimal control is proved. An optimal control characterization in terms of state and adjoint functions is provided. Furthermore, a second condition of optimality is given. The optimality systems are solved based on an iterative discrete scheme that converges following an appropriate test similar the one related to the forward–backward sweep method. Numerical results are provided to illustrate the effectiveness of our approach for several scenarios.

/pdf/an-optimal-control-problem-for-a-spatiotemporal-sir-model-20tx1r590g.pdf

An optimal control problem for a spatiotemporal SIR model

In the present work, we consider a mathe- matical model of an SIR epidemic model with saturated incidence rate and saturated treatment function. We use an optimal vaccination strategies to minimize the susceptible and infected individuals and to maximize the number of recovered individuals. We work in the nonlinear optimal control framework. Some results concerning the existence and the characterization of the optimal control will be given. Numerical simulations are also presented.

/pdf/optimal-vaccination-strategies-of-an-sir-epidemic-model-with-g906zu02vt.pdf

Optimal Vaccination Strategies of an SIR Epidemic Model with a Saturated Treatment

We will investigate the optimal control strategy of an SIR epidemic model with time delay in state and control variables. We use a vaccination program to minimize the number of susceptible and infected individuals and to maximize the number of recovered individuals. Existence for the optimal control is established; Pontryagin’s maximum principle is used to characterize this optimal control, and the optimality system is solved by a discretization method based on the forward and backward difference approximations. The numerical simulation is carried out using data regarding the course of influenza A (H1N1) in Morocco. The obtained results confirm the performance of the optimization strategy.

/pdf/optimal-control-of-an-sir-model-with-delay-in-state-and-2xwc7race4.pdf

Mostafa Rachik

Papers

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

Optimal control of an epidemic model with a saturated incidence rate

An optimal control problem for a spatiotemporal SIR model

Optimal Vaccination Strategies of an SIR Epidemic Model with a Saturated Treatment

Optimal Control of an SIR Model with Delay in State and Control Variables