Optimal Control with Treatment and Water Sanitation for Cholera Epidemic Model

. This paper proposes a mathematical model for cholera using optimal control of treatment through quarantine and water sanitation. Cholera is acute diarrhoea caused by Vibrio cholera bacteria infecting the intestinal tract. The analysis related to the spread of this disease is carried out through a mathematical approach. The constructed mathematical model is demonstrated epidemiologically. The proposed optimal control is the treatment of infected individuals during the quarantine period and sanitation, namely environmental hygiene, especially water. This strategy aims to suppress the number of infected individuals and reduce the concentration of bacteria due to cholera disease. To solve the optimal control problem, we employ the 4th-order forward-backward Runge-Kutta method. Based on the simulation results, the number of individuals infected by cholera and the concentration of bacteria decreased significantly. Moreover, the proposed method can transfer infected to recovered individuals faster than an optimal control treatment.


Optimal Control Solution
The 4th-order forward-backwards Runge-Kutta method is employed to address the optimal control issue.

Numerical Simulation
In general, numerical simulations help test mathematical models constructed based on the object or problem under study. For example, numerical simulations can be used in optimal control to analyze how effective the proposed control strategy is. The simulation is carried out using the MATLAB program.

Drawing Conclusions and Suggestions
Based on the analysis of the simulation results that have been carried out, a conclusion can be drawn on the proposed strategy and suggestions for future research.

Results and Discussion
In this section, the dynamic model of cholera transmission in SIQRB type is modified by including treatment methods for individuals in the quarantine period and water sanitation category. The subsequent phase involves determining the optimal control approach for the model established and evaluated through numerical simulation. We utilize cholera epidemic data from the Artibonite Department of Haiti between November 2010 and May 2011 [26] as in [2].

Mathematical Model Formulation of Cholera
This section proposes a type of SIQRB model that considers the concentration of bacteria in the dynamic cholera model. At any given time ≥ 0, the overall population ( ) is clustered into 4 clusters, namely individuals susceptible to cholera infection, ( ), individuals infected by cholera displaying symptoms, ( ), individuals under treatment via quarantine, ( ), and individuals declared recovered from cholera, ( ). On the other hand, a cluster of bacteria, ( ), is also Heri Purnawan, Rifky Aisyatul Faroh, et al.
considered, reflecting the bacteria concentration at time-. The initial conditions and parameters used in the SIQRB model can be shown in Table 1 [2], [3].
This study proposes optimal control, namely a treatment for infected individuals by cholera through quarantine and water sanitation, into the dynamic cholera model. The results of model construction by considering this strategy can be represented by a system of nonlinear ordinary differential equations as follows: where 1 ( ) is the treatment for cholera-infected individuals through quarantine, with 0 ≤ 1 ( ) ≤ 1 [2], and 2 ( ) is water sanitation, with 0 ≤ 2 ( ) ≤ 0.1 [24]. Equation (1) is a dynamical system of cholera spreading model modified from [2], [3] by considering water sanitation as the additional optimal control strategy.

Optimal Control Solution
The objective function of the optimal control issue with treatment for cholera-infected individuals through quarantine and water sanitation can be defined as follows: with transversality conditions * ( ) = 0, = 1,2, ⋯ , 5

Numerical Simulation
In this section, numerical simulations of optimal control problems are carried out in Equation (2) for = 2000 [2] so that 1 = 1000 while 2 = 20 [24]. The 4th-order forward Runge-Kutta method is used to solve the state equations, while the solution to the co-state equations is used by the 4th-order backward Runge-Kutta method [28], [30]. In this simulation, the parameters used can be seen in Table  1. To compare the simulation results on individuals infected by cholera and the concentration of bacteria with and without control, the sub-model from Equation (1) can be written as follows [2], [3]: where the sub-model is an interpretation of the cholera mathematical model without optimal control and quarantine which assumes that = = = 2 = (0) = (0) = 0. A comparison of submodel simulations, optimal control strategy with treatment ( 1 ) and optimal control strategy with treatment ( 1 ) and water sanitation ( 2 ) can be seen in Figure 2. In contrast, simulations for optimal controls 1 and 2 can be seen in Figure 3.

Figure 2.
Optimal solutions of * and * with treatment and water sanitation strategies based on initial conditions and parameters presented in Table 1 Figure 2 shows the simulation results of the infectious individuals and the concentration of bacteria with and without control. The simulation results of the cholera sub-model (without control) and with control, namely treatment are obtained from [2], whereas the simulation results with treatment and water sanitation display our proposed strategies. For the application of one optimal control, individuals infected with cholera which was initially around 4500, became about 3300, whereas, with two optimal controls at the same time, it could suppress individuals infected by cholera which was initially about 4500 to around 3100, as shown in Figure 2.
According to Figure 3, the optimal control 1 yields the highest possible value for ∈ [0, 110] days. For ∈ [110, 182], optimal control 1 experiences a decline in its endpoint, whereas optimal control 2 reaches its peak value for ∈ [0, 182] days. The maximum value of optimal input 2 indicates a water sanitation strategy must be carried out continuously. Consequently, practical methods for controlling cholera, such as treatment ( 1 ) and water sanitation ( 2 ), have reduced the number of individuals affected by the disease. The difference in applying only treatment and the combination between treatment and water sanitation occurs when = 15 days.
Heri Purnawan, Rifky Aisyatul Faroh, et al.  Table 1 Based on the simulation results, optimal control 1 * and 2 * allows a significant decrease for the number of individuals infected by cholera and the concentration of bacteria, as illustrated in Figure 2. When the controls are implemented, there is a significant drop in the highest number of individuals infected by cholera. In addition, the suggested measures to control cholera can facilitate the transfer of the disease from infected individuals to those who have recovered.

Conclusion
This study proposes a strategy for controlling cholera outbreaks by treating individuals infected by cholera through quarantine and water sanitation. Based on the highlighted simulation results previously, it can be concluded that applying one optimal control, namely treatment and the combination of two optimal controls, namely treatment and water sanitation, gives a significant reduction in the number of individuals infected by cholera as well as in the concentration of the bacteria that causes the disease. Therefore, the authors propose that implementing further optimal strategies, such as vaccinating susceptible individuals against cholera using the SIQRB model, could lead to a more optimal.