Optimal control of the lost to follow up in a tuberculosis model

Abstract

This paper deals with the problem of optimal control for the transmission dynamics of tuberculosis (TB). A TB model that considers the existence of a new class (mainly in the African context) is considered: the lost to follow up individuals. Based on the model formulated and studied in the work of Plaire Tchinda Mouofo, (2009), the TB control is formulated and solved as an optimal control theory problem using the Pontryagin's maximum principle (Pontryagin et al., 1992). This control strategy indicates how the control of the lost to follow up class can considerably influence the basic reproduction ratio so as to reduce the number of lost to follow up. Numerical results show the performance of the optimization strategy.

Figure 1

Flow diagram of the model without control.

Figure 2

Flow diagram of the model with control.

Figure 3

Variation of the basic reproduction ratio without control as a function of β, with ϕ = 0.0022 and k ₂ = 0.0006.

Figure 4

The influence of the control with S(0) = 50, E(0) = 100, I(0) = 150, L(0) = 200, β = 0.002, ϕ = 0.0022, and k ₂ = 0.0006. All the other parameter values are as in Table 1.

Figure 5

The influence of the control with S(0) = 50, E(0) = 100, I(0) = 150, L(0) = 200, β = 0.003, ϕ = 0.1, and k ₂ = 0.0006. All the other parameter values are as in Table 1.

Figure 6

The influence of the control with S(0) = 50, E(0) = 100, I(0) = 150, L(0) = 200, β = 0.02, ϕ = 0.5, and k ₂ = 0.0006. All the other parameter values are as in Table 1.

Figure 7

The influence of the control with S(0) = 50, E(0) = 100, I(0) = 150, L(0) = 200, β = 0.02, ϕ = 0.5, and k ₂ = 0.006. All the other parameter values are as in Table 1.