19] analyzed a model of the spread of AAT in livestock considering insecticide use. Sensitivity analysis and autonomous simulation to determine the resilience of each parameter in our proposed model to the dynamics of rhinoceros and tsetse fly populations are discussed in Section 5 and also the basic reproduction number. The construction of the mathematical model for the optimal control problem is done in Section 6.
The characterization, numerical simulations, and cost-effectiveness of the optimal control problem are also discussed in Section 6. In addition, System (1) is satisfied by the following non-negative initial condition:. 2) The following theorem guarantees that System (1) always has a unique solution and each solution is always non-negative as long as the initial condition is also non-negative. In many epidemiological models, the baseline reproduction number determines the presence or absence of the disease.
In many epidemiological models involving species interaction, basic reproduction plays an essential role in determining the existence of the disease. However, in some conditions, a basic reproductive number below one does not ensure the disappearance of the disease. Therefore, we know that the multiplication of the root of 𝐹(𝑉) will be negative if ℛ0 > 1.
Based on the analysis in this section, it appears that the existence of a base reproduction number less than one as an indicator of the disappearance of AAT can no longer always be guaranteed. This is because there is still a chance for the existence of AAT endemic equilibrium points even if the value of the basic reproduction number is less than one. Increasing the value of 𝛽𝑟2 will increase the chance of the occurrence of backward bifurcation in our model.
From the previous section, we see in Theorem 5 the possibility of the existence of two AAT-endemic equilibria for a value of ℛ0< 1. For many classical models of disease transmission [34–36], the condition of a smaller basic reproduction number that one is enough to guarantee the disappearance of the disease. Based on the results of the elasticity analysis using the normalized forward elasticity analysis method, we next show the dynamics of the solution of System (1) for the total populations of infected rhinoceros and infected flies for variations in the values 𝑢1 and 𝑢2.
In addition, it can also be seen that spraying the ground with sufficient intensity has been able to prevent outbreaks in the number of infections among the rhinoceros and fly populations. The results of the ℛ0 sensitivity analysis show that early detection and ground spraying have the potential to reduce the spread of AAT disease. However, the high intensity of the intervention results in high implementation costs in the field.
Based on this, the reconstruction of the model in System (1) to an optimal control model will be discussed in this section.
Characterization of the Optimal Control Problem
For our model we choose a quadratic cost function in our objective function, which is common in epidemiological models [45–48]. Our optimal control problem is to find some controls 𝑢1∗ and 𝑢2∗ corresponding to the state variable 𝑋∗ in the interval [0, 𝑇]. In our case, the problem is to find 𝑢1∗ and 𝑢2∗ that minimize the cost function (24) corresponding to the state variable 𝑋∗.
We derive the necessary condition for our optimal control problem using the well-known maximum principle of Pontryagin [22,49]. Due to the limited ability to implement the control variables, we chose that each control variables should be in the interval of [0, 1]. To summarize, our optimal control problem consists of the state system (23) with a non-negative initial state (2), additional system (26) with a transversality condition 𝜆𝑖(𝑇) = 0 for 𝑖 = 1,2, .
Numerical Experiments
Different combinations of interventions
As explained in the previous subsection, both interventions succeed in suppressing the number of infected rhinoceroses and flies almost throughout the entire simulation, with the exception of the number of infected rhinoceroses detected. This glitch occurred because of the high intensity of early detection at the beginning of the simulation. It is clear that when the policy makers rely solely on infection detection to prevent the spread of AAT, the outcome is not as good as in the first scenario.
However, the intervention cost for the second scenario was much lower than for the first scenario, ie, only 5,781.9, almost twice that of the first scenario. In the last simulation, which included the third scenario, soil spraying was the only intervention implemented. It can be seen that the control trajectories of 𝑢2 in the third scenario were somewhat similar to the first scenario, i.e., high intensity should be applied at the beginning and decrease when the number of infected flies starts to decrease.
As a result, the number of infected rhinos and flies decreased much more compared to the second scenario, but not as much as in the first scenario. From the numerical experiments with all the above scenarios it can be seen that all scenarios provided a reduced number of infected rhinos and flies. The second method used to analyze the cost-effectiveness of the above strategies was the Incremental Cost-Effectiveness Ratio (ICER) method.
Since ICER-1 > ICER-3, we can conclude that the third strategy, i.e. implementing soil spraying as a single intervention is the best strategy. A large number of reports have noted the emergence of African Animal Trypanosomiasis (AAT) in the near-endangered white rhinoceros population. Sensitivity analysis of the basic reproduction number showed that infection detection and soil spraying both have the potential to suppress the spread of AAT.
From numerical simulations of the optimal control problem, it was found that time-dependent interventions successfully reduced the spread of AAT in the white rhinoceros and tsetse fly populations. 2] Food and Agriculture Organization of the United Nations, Guidelines on Good Practices for Land Application of Pesticides (2001). Rabinowitz, A mathematical model for evaluating the role of trypanocide treatment of cattle in the epidemiology and control of trypanosoma brucei rhodesiense and t.
Metz, On the definition and calculation of the basic reproduction ratio R0 in infectious disease models in heterogeneous populations, Journal of Mathematical Biology. Cushing, Determining important parameters in the spread of malaria using sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology.
