Different combinations of interventions

6.2 Numerical Experiments

6.2.2 Different combinations of interventions

Unlike in the previous subsection, numerical scenarios will be carried out based on the type of combination of interventions used in this subsection. The scenarios are divided into three, namely when both interventions are given (first scenario), when infection detection is a single intervention (second scenario), and when ground spraying is a single intervention (third scenario). To conduct the simulation in this section, we used initial condition IC1 for all scenarios.

Figure 7 shows the numerical result for the first scenario. As already explained in the previous subsection, both interventions succeed in suppressing the number of infected rhinos and flies almost during the whole simulation, except for the number of detected infected rhinos. This glitch occurred due to the high intensity of early detection at the beginning of the simulation.

The result for the second strategy is given in Figure 9. It can be seen that when the policymakers only rely on infection detection to prevent the spread of AAT, the result is not as good as in the first scenario. It can be seen that the reduced

number of infected rhinos and flies was not as high as in the first scenario.

However, the cost of intervention for the second scenario was much lower than for the first scenario, i.e., only 5,781.9, almost twice smaller than in the first scenario.

Figure 9 White rhinos, flies and control trajectories under the intervention strategy of the second scenario (𝑢₁(𝑡) ≥ 0, 𝑢₂(𝑡) = 0).

In the last simulation, which involved the third scenario, ground spraying was the only intervention implemented. The results are given in Figure 10. It can be seen that the control trajectories of 𝑢₂ in the third scenario were somewhat similar with the first scenario, i.e., high intensity should be implemented at the beginning and decrease when the number of infected flies starts decreasing. 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. The cost of intervention for the third scenario was 6,320.1.

Figure 10 White rhinos, tsetse flies and control trajectories under the intervention strategy of the third scenario (𝑢1(𝑡) = 0, 𝑢₂(𝑡) ≥ 0).

Cost-effectiveness analysis. From the numerical experiments with all of the above scenarios, it can be seen that all scenarios provided a reduced number of infected rhinos and flies. However, a good result comes with high intervention costs. Hence, we have to determine which is the best strategy compared to the other in terms of cost-effectiveness. To do this, we used two types of cost- effectiveness measures, namely the Average Cost-Effectiveness Ratio (ACER) and the Incremental Cost-Effectiveness Ratio (ICER).

The ACER method aims to determine which strategy produces the lowest average cost for each reduction of one infected individual. The formula for calculating the ACER value is as follows:

𝐴𝐶𝐸𝑅 = 𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 𝑓𝑜𝑟 𝑖𝑛𝑡𝑒𝑟𝑣𝑒𝑛𝑡𝑖𝑜𝑛

𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑣𝑒𝑟𝑡𝑒𝑑 (30) A lower ACER value indicates a better result. A low ACER value means that the average cost for each infected rhino averted is also small. The result of the ACER analysis is given in Figure 11. From Figure 11, we can see that intervention of

ground spraying as a single intervention was more cost-effective compared to the others, followed by the first and the second scenario, respectively.

Figure 11 The result of ACER analysis.

The second method used to analyze the cost-effectiveness of the above strategies was the Incremental Cost-Effectiveness Ratio (ICER) method. This method aims to find the best strategy from two compared strategies. The following equation expresses the formula:

𝐼𝐶𝐸𝑅_𝑖−𝑗 = 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑐𝑜𝑠𝑡 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑦−𝑖 𝑎𝑛𝑑 𝑗

𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 # 𝑜𝑓 𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑣𝑒𝑟𝑡𝑒𝑑 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑦−𝑖 𝑎𝑛𝑑 𝑗 (31) First, we rank our strategies from the smallest number of infected rhinos averted to the highest number of infected rhinos averted for each strategy. The result is given in Table 3.

Table 3 All scenario in increasing order based on the number of rhino infections averted.

Strategy Total infections

averted Total cost ICER

2^nd scenario 32 210.2 5 781.9 0.1795

3^rd scenario 115 489.7 6 320.1 0.00646

1^st scenario 131 833.9 10 721.3 0.269

Since ICER-2 > ICER-3, we eliminate the second scenario for the next calculation. Hence, we can conclude that the second scenario was the least cost- effective strategy based on the ICER analysis. For the next calculation, we compare the third with the first scenario. The result is given in Table 4. Since ICER-1 > ICER-3, we can conclude that the third strategy, i.e., the implementation of ground spraying as a single intervention, is the best strategy.

This result is in line with our previous analysis using the ACER method.

Table 4 Comparison between third and first scenarios Strategy Total infection averted Total cost ICER

3^rd scenario 115 489.7 6 320.1 0.05

1^st scenario 131 833.9 10 721.3 0.269

7 Conclusion

A large number of reports have noted the emergence of African Animal Trypanosomiasis (AAT) in the Nearly Threatened white rhino population. Thus, qualitative research related to efforts to minimize the spread of AAT is important, requiring serious attention from various parties, including mathematicians.

Therefore, this article introduced a mathematical model for AAT, involving populations of white rhinos and tsetse flies, and two interventions, namely infection detection and ground spraying. Mathematical model analysis showed how the AAT-free equilibrium point is always locally asymptotically stable when the basic reproduction number is less than one. On the other hand, it is possible to have multiple AAT-endemic equilibriums when the basic reproduction number is less than one. A unique AAT-endemic equilibrium always exists when the basic reproduction number is larger than one. Bifurcation analysis using the Castillo-Song theorem [21], showed the possibility of backward bifurcation emerging from our model. Hence, there is a situation when the basic reproduction number being less than one is not enough to guarantee the disappearance of AAT from the population [37-40]. Sensitivity analysis of the basic reproduction number showed that infection detection and ground spraying both have potential to suppress the spread of AAT. However, we found that ground spraying is more

effective in influencing the basic reproduction number’s size than infection detection.

From numerical simulations of the optimal control problem, it was found that time-dependent interventions successfully reduced the spread of AAT in the white rhino and tsetse fly populations. The cost-effectiveness analysis found that ground spraying as a single intervention is a better strategy than infection detection as a single intervention or a combination between infection detection and ground spraying. In addition, we also found that preventive interventions are more advisable to save on intervention costs rather than waiting for a high number of infections and then implementing interventions.

This research showed that AAT can significantly influence the dynamics of white rhino populations. To suppress the spread of AAT, ground spraying is highly recommended as a single intervention to prevent possible endemics.

Acknowledgement

This research was financially supported by RistekBRIN, Indonesia through the PDUPT Research Grant Scheme 2021 (ID number: NKB- 165/UN2.RST/HKP.05.00/2021).

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