Sensitivity analysis of the basic reproduction number give information of the parameters which are sensitive to changes in the basic reproduction number [8]. Such parameters form the best candidates for control. This study utilized the normalized forward sensitivity index which was defined as

where the basic reproduction number was calculated using the next generation matrix and explained in [9]; and obtained as

where k 1 = ( μ + η c + η + τ ) k 2 = ( ε + μ + δ d ) k 3 = ( κ + μ + σ d + ε c + γ ) k 4 = ( μ + λ + δ c ) k 5 = ( ρ c + μ + σ m + α ) k 6 = ( μ + δ c + α c ) k 7 = ( ξ 1 + ξ 2 + μ ) k 8 = μ v k 9 = μ v The spread dynamics of malaria and dengue fever were further analyzed numerically to determine the spread dynamics patterns. The numerical values of the parameters were derived from secondary sources and were summarized in the table below.

Table 1. Numerical values for malaria-dengue co-infection model parameters.

This section focused on determining the most sensitive parameter of the basic reproduction number. It utilised the numerical values of the simulation described in Table 1 above. The sensitivity indices were summarized in the sensitivity graph below. The positive indices implies that such parameters have a direct proportionality with the basic reproduction number [23], and are coloured navy blue in the graph. The negative sensitivity indices imply that such parameters have an inverse proportionality with the basic reproduction number and are coloured maroon in the graph in Figure 2 below.

Figure 2. Sensitivity analysis of malaria dengue coinfection spread dynamics.

The parameters with moderate positive sensitivity include the rate of misdiagnosis of dengue infected patients, and the rate at which patients in the incubation period confuse dengue with malaria. Therefore, these two parameters also form good candidates for control in the control measures. The parameters with the most negative sensitivity indices include the rate at which misdiagnosis is corrected and the natural death rate of human beings. Therefore, the control measures should target these parameters to enhance them so as to contain the spread of dengue and malaria. These parameters formed the basis of the optimal control solution where control measures such as retooling of health care providers u 1 , public sensitization on how to prevent the two mosquito-borne diseases u 1 , and administration of antimalarial drugs u 1 as the some of the key control measures that could contain the spread of the disease.

The pontryagin maximum principle served as the bridge between the spread dynamics and the optimized control of the coinfection by introducing the Hamiltonian [27][28]. It converts the system of differential equation in Equation (1) and Equation (6) into a minimizing point-wise problem using the Hamiltonian ℍ . The Hamiltonian of the problem was defined as

Equation (8) was expanded to

where λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , λ 7 , λ 8 , λ 9 , λ 10 are the adjoint variables associated with each compartment. The optimality of the coinfection requires that

where the function x ( t ) = ( S , E d , M d , I d , R d , E m , I m , R m , I c , R c , t ) . The result of the calculations was obtained as

To solve the system of differential equations for optimality we needed to determine the transversality conditions which give the terminal conditions at t = T . The transversality conditions were given as

The condition in Equation (25) guarantees the existence of an optimal solution that encompass control variables u 1 * , u 3 * , u 3 * that justifies the functional J ( u 1 * , u 2 * , u 3 * ) = min u 1 , u 2 , u 3 { J ( u 1 , u 2 , u 3 ) : u 1 , u 2 , u 3 ∈ U } . The stationary points of the hamiltonian with to control variables give the extremal points that fulfill the Pontryagin’s maximum principle. The function of the formula was defined as

The result of the differentiation with respect to u 1 is given by

Making u 1 the subject to obtain the optimal control characterization at u 1 = u 1 * , we obtained

The process was repeated again by differentiating the hamilitonian wit respect to u 2 to obtain the optimal control characterization at u 2 = u 2 * and obtained. The differentiation of the hamilitonian with respect to u 2 was obtained as

The optimal control characterization was obtained by making u 2 the subject of the equation at the point u 2 = u 2 * and the result was obtained as

The third optimal control characterization was obtained by differentiating the hamiltonian with respect to u 3 and obtained

making u 3 the subject of the equation and obtaining the optimal control characterization at u 3 = u 3 * as

Theorem

Considering the state variable solutions given by S ** , E d ** , M d ** , I d ** , R d ** , E m ** , I m ** , R m ** , I c ** , R c ** , S v ** , E v ** , I v m ** , I v d ** whose optimal solutions are contained in the control set U * = ( u 1 * , u 2 * , u 3 * ) which are the optimized solution of the optimal control problem 1. The costate functions are given by λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , λ 7 , λ 8 , λ 9 , λ 10 , λ 11 , λ 12 , λ 13 , λ 14 which are the solutions of the Equations (11)-(24). The full characterization equation of the optimization u 1 * , u 2 * , u 3 * together with bonds was given by

Proof

The existence of an optimal control solution of 1 over the control and convex set U = u 1 * , u 2 * , u 3 * is guaranteed by the convexity of the integral of the functional J ( u 1 * , u 2 * , u 3 * ) . The state condition are satisfied by the boundedness that existed prior leading to the satisfaction of the Lipschitz property of the state system. The boundedness was described as

whose boudedness was given as follows

To establish the uniqueness of the optimal control solution, traditional analytical frameworks can be employed as long as the simulation timeframes remain sufficiently small. This specific methodology was adopted for this study because both the state and costate equations are characterized by Lipschitz continuous functions. Furthermore, since the optimal control system is inherently bounded, the costate equations utilized linear bounded coefficients, which ensured that the costate variables themselves possessed defined upper bounds. Restricting the duration of the time interval served as a vital condition to formally guarantee that the system’s optimal control is unique. This small-time requirement is fundamentally driven by the opposing temporal directions of the optimality system; while the state problem is defined by initial conditions and solved forward, the adjoint problem is defined by terminal conditions and solved backward, creating a complex two-point boundary value structure.

The section focused on presenting a numerical solution of the optimal control problem when compared with the non-optimized problem. Forward iteration method was utilized to solve the optimal control differential equations in Equation (1). The system of differential equation 1 incorporates the control measures u 1 , u 2 and u 3 . The backward iteration technique was used to solve the co-state system of differential equations with the control bounds and initial conditions as the boundary conditions. The controls variables were updated progressively using the weighted convex combination that supported both backward and forward iteration until the optimum solution was reached. The numerical scheme was implemented in the maple software by utilizing both runge-kutta order four method and the forward-backward sweep methods. The numerical values of the parameters were taken from the numerical values listed in Table 1.

The initial conditions of the optimal control problem were taken at the 100th day of the general spread dynamics [9]. At this point, E d ( t ) = 12384.258970 , E m ( t ) = 3132.981741 , I c ( t ) = 1218.741262 , I d ( t ) = 11536.904329 , I m ( t ) = 3659.732227 , M d ( t ) = 3676.038969 , R c ( t ) = 8138.870086 , R d ( t ) = 7689.097503 , R m ( t ) = 1298.592250 , S ( t ) = 47251.178561 [29]. To contain the spread of malaria and dengue fever, the number of exposed, infectious and coinfected compartments has to be minimized. The relative importance of the compartments to be minimized was assumed to be equal, that is A 1 = A 2 = A 3 = A 4 = 1 . The unit cost of retooling healthcare workers was taken to be 200 shillings thus the weighting C 1 = 200 [29], the unit cost public sensitization was taken to be 100 shillings thus the weighting C 2 = 100 , and the unit cost of treating malaria through administering antimalaria drugs was taken to be 800 shillings thus making the weighting component C 3 = 800 [30]. The above numerical values were incorporated into the following strategies of implementing the control parameters.

1. Strategy 1: The implementation of retooling and public sensitization, that is u 1 ≠ 0 , u 2 ≠ 0 , and u 3 = 0 .

2. Strategy 2: The administration of antimalarial drugs and retooling of health care workers, that is u 1 ≠ 0 , u 2 = 0 , and u 3 ≠ 0 .

3. Strategy 3: The implementation of all the three strategies, that is, retooling heath care workers, conducting public sensitization campaigns and administering the antimalarial drugs i.e u 1 ≠ 0 , u 2 ≠ 0 , and u 3 ≠ 0 .

This strategy reduces the misdiagnosed population progressively from the first day implementing the control measures. The control measures stabilizes at 4500 people per day while the uncontrolled disease dynamics increase exponentially as illustrated in Figure 3(a). Figure 3(b) illustrates the impact of application of the two control values, the implementation of the control variables reduces the number of infected individuals daily.The strategy also reduced the infected malaria population limiting the numbers below 4000 effectively creating an upper limit to the spread dynamics of the disease under controlled environment. In contrast, the the uncontrolled spread dynamics of individuals with malaria increase consistently leading to uncontrolled spread of the disease as illustrated in Figure 3(c). The coinfected spread dynamics also show robust response to the control measures by breaking the increasing population and flattening the curve as illustrated in Figure 3(d).

This strategy targets training the of workers to better equip and the administration of antimalarial drugs. The combination of the two control measures in this strategy are illustrated in Figure 4. The strategy takes time before gaining traction in controlling the number of misdiagnosed people per day. The strategy fails to show any meaningful difference between the controlled spread dynamics and the uncontrolled spread dynamics in the first 15 days of the simulation. However, there is a marked difference between the controlled and uncontrolled spread dynamics. The strategy has little impact on the spread dynamics as shown in Figure 4(a). The dengue fever spread dynamics show an increasing in the controlled spread dynamics when compared to the uncontrolled spread dynamics as shown in Figure 4(b). This unconventional behaviour rises from focus of the strategy on the control of malaria at the expense of dengue fever. The number of people infected with malaria under this strategy shows progressive decline to values near zero as shown in Figure 4(c). The impact of the strategy on the coinfected people per day was effective in reducing the number of coinfected people to a minimum of around 200 people per people in the first 15 days of the control simulation. The number of coinfected people per day starts to increase slowly after the first 15 days but still remains in low levels. The uncontrolled spread dynamics remains high compared to the controlled spread dynamics as shown in Figure 4(d) below.

Figure 3. Comparative graphs of different infectious groups under strategy 1 optimal control.

Figure 4. Comparative graphs of different infectious groups under strategy 3 optimal control.

This strategy is a combination of all the controlled variables under consideration; that is retooling of human resources of health care providers, conducting public sensitization campaigns, and administration of antimalarial drugs. The impact of the spread dynamics is illustrated in Figure 5 below.

Figure 5(a) illustrates the spread dynamics of misdiagnosed population under the application of the three control variables. The control variables have an immediate impact on the people misdiagnosed with dengue with the control spread dynamics having lower levels and the uncontrolled spread dynamics having high values. The impact of the strategy on the compartment of infected dengue population starts immediately leading to the decrease to the number infected people with dengue fever as illustrated in Figure 5(b). The impact of the strategy on the people infected with malaria is illustrated in Figure 5(c) below. The number of infected people with malaria decreases drastically to numbers near zero within the

Figure 5.Comparative graphs of different infectious groups under strategy 4 optimal control.

70 days of simulation when the simulation is under control. On the other hand, the uncontrolled spread dynamics remain high when compared to the controlled spread dynamics as shown in Figure 5(c). The spread dynamics of coinfected people with malaria and dengue fever under control is illustrated in Figure 5(d). The number of people coinfected people per day decreases rapidly in the first 10 days of the simulation to around 200 people per day. The number of coinfected people starts to gradually increase after the 15th day to a maximum of about 400 people per day. The uncontrolled spread dynamics increase consistently to values of about 1900 people per day. This strategy is the most effective since it guarantees a reduction in all the infected compartments of the spread dynamics of malaria and dengue fever.