Date Published: February 6, 2017
Publisher: Public Library of Science
Author(s): Folashade B. Agusto, Ibrahim M. ELmojtaba, Henk D. F. H. Schallig.
In this paper, a deterministic model involving the transmission dynamics of malaria/visceral leishmaniasis co-infection is presented and studied. Optimal control theory is then applied to investigate the optimal strategies for curtailing the spread of the diseases using the use of personal protection, indoor residual spraying and culling of infected reservoirs as the system control variables. Various combination strategies were examined so as to investigate the impact of the controls on the spread of the disease. And we investigated the most cost-effective strategy of all the control strategies using three approaches, the infection averted ratio (IAR), the average cost-effectiveness ratio (ACER) and incremental cost-effectiveness ratio (ICER). Our results show that the implementation of the strategy combining all the time dependent control variables is the most cost-effective control strategy. This result is further emphasized by using the results obtained from the cost objective functional, the ACER, and the ICER.
Malaria and visceral leishmaniasis (VL) are two major parasitic diseases with overlapping distributions which are both epidemiological and geographical in nature. This overlap may consequently lead to co-infection of the two parasites in the same patients . Due to this co-infection, these parasites may partially share the same host tissues, with the ability to evade and subvert the host immune response; the clinical outcomes, however, depend largely on the immunological status of the host . Furthermore, the success of the visceral Leishmania donovani complex obligate intracellular parasites in colonizing the macrophages and other reticulo-endothelial cells of the lymphoid system is due to their ability to alter the host’s parasite destruction signaling pathways and adaptive immunity engagement .
In this study, we consider the model without control proposed and analyzed by Elmojtaba . The model examined the dynamics of the malaria and visceral leishmaniasis co-infection in four populations; human host population Nh(t), reservoir host population Nr(t), mosquito population Nvm(t), and sandfly population Nvl(t). The human host population was divided into eight categories, individuals susceptible to both malaria and visceral leishmaniasis Sh(t), those who are infected with malaria only Ihm(t), those who are infected with visceral leishmaniasis only Ihl(t), those who are infected with both malaria and visceral leishmaniasis Ihml(t), The population also include those who have developed post kala-azar dermal leishmaniasis (PKDL) after the treatment of visceral leishmaniasis Ph(t), those who have developed PKDL and have malaria Phm(t), those who are recovered from leishmaniasis and have permanent immunity but susceptible to malaria Rh(t) and those who are recovered from leishmaniasis and infected with malaria Rhm. Hence, the total human population is given as
Following [13, 14] we used the normalized forward sensitivity index also called elasticity, as it is the backbone of nearly all other sensitivity analysis techniques  and are computationally efficient . The normalized forward sensitivity index of the quantity Q with respect to the parameter θ is given by:
Using the elasticity formula (2) and the parameter sets in Table 3, we now obtain numerical values for the elasticities. For each parameter θ we calculated the elasticity index of R′c with respect to θ. Results are displayed in Table 4
Following the conclusion obtained from the sensitivity analysis, we introduce into the malaria-visceral leishmaniasis model (1) four time-dependent controls u1(t), u2(t), u3(t) and u4(t). These time-dependent controls represent the use of personal protection measures (u1(t) and u2(t)) such as the use of insecticide-treated nets, application of repellents or insecticides to skin or to fabrics and impregnated animal collars (particularly dogs)  and the use of windows and door screens to prevent both mosquitoes and sandflies bites both during the day and at night. Furthermore, the time-dependent control u3(t) represents the culling of infected reservoir animals (like dogs) and the control u4(t) represents the use of insecticides such as DDT, pyrethroids and residual spraying of dwellings and animal shelters  to kill the mosquitoes and sandflies. Thus, the malaria-visceral leishmaniasis model (1) with time-dependent control is given as:
Thus, we want to find the optimal values (u1*,u2*,u3* and u4*) that minimizes the cost objective functional J(u1, u2, u3, u4) where
This performance specification involves the numbers of infected humans, reservoirs, mosquitoes and sandflies, along with the cost of applying the controls (u1(t), u2(t), u3(t) and u4(t)). The coefficients, Ai, Cj, i = 1⋯8, j = 1⋯4, are balancing cost factors and tf is the final time. The control quadruple (u1(t), u2(t), u3(t) and u4(t)) are bounded, Lebesgue integrable functions [28, 29]. The goal is to find the optimal control, u1*,u2*,u3* and u4*, such that
where the control set,
Numerical solutions to the optimality system comprising of the state eq (3), adjoint eq (7), control characterizations Eq (8) and corresponding initial/final conditions are carried out using the forward-backward sweep method (implemented in MATLAB) and using parameters set in Table 3. The algorithm starts with an initial guess for the optimal controls and the state variables are then solved forward in time using Runge Kutta method of the fourth order. Then the state variables and initial control guess are used to solve the adjoint equations Eq (7) backward in time with given final condition (8), employing the backward fourth order Runge Kutta method. The controls u1(t), u2(t), u3(t), u4(t) are then updated and used to solve the state and then the adjoint system. This iterative process terminates when the current state, adjoint, and control values converge sufficiently .
Next, we performed a cost-effectiveness analysis. In order to justify the costs associated with health intervention(s) or strategy (strategies) such as treatment, screening, vaccination or educational intervention, the associated benefits are usually evaluated using cost-effectiveness analysis . In this section we will consider three approaches, the infection averted ratio (IAR), the average cost-effectiveness ratio (ACER) and the incremental cost-effectiveness ratio (ICER).
In this paper, we applied optimal control theory to malaria/visceral leishmaniasis co-infection model developed in . The analysis shows that the disease-free equilibrium of the model is locally asymptotically stable whenever the associated reproduction number (R0), is less than unity and unstable otherwise. The model also exhibits backward bifurcation, a phenomenon where two stable equilibria coexist when the reproduction number is less than unity.