## 3 Results  
### From Disease Persistence to Elimination in a Scenario of Increasing Heterogeneity  
#### 3.1.1 Heterogeneity and Ivermectin  
The present model was implemented for sets of ten simulations in three heterogeneity scenarios, defined by the following values of heterogeneity affinity: low (\(\theta=1,0.66\)), intermediate (\(\theta=0.5,0.4\)), and strong (\(\theta=0.33,0.29,0.25\)). See Equation (1). These values are equivalent to the set of PW-rules of 20/20, 20/30, 20/40, 20/50, 20/60, 20/70, and 20/80, respectively. Our results show a consistent linear correlation between 30 year average malaria incidence and increasing levels of heterogeneity, suggesting that heterogeneity strongly promotes malaria transmission (Figure 2).  
Next, we analyze how heterogeneity in disease transmission is influenced by ivermectin treatment, i.e., by the level of drug administration. In the absence of ivermectin administration, malaria incidence is three times higher in high heterogeneity simulations related to the PW-rule of 20/80 (\(\theta=0.25\)), for which around 150 annual cases pHHI are observed, when compared to simulations in full homogeneity (\(\theta=1.0\)) where one observes \(\sim\)50 annual cases pHHI. The curve in Figure 3a for \(p_{iv}=0\) clearly shows this pattern. Moreover, this pattern is significantly reduced if ivermectin is used by even a small fraction of 5% of the general population, and a major decline in malaria incidence, very close to disease extinction, can be achieved by using ivermectin treatment in 20% of the general population, in the high heterogeneity scenario (PW-rule of 20/80, \(\theta=0.25\)).  
An important feature is revealed in Figure 3b: isolines of constant malaria incidence fit approximately a linear relation \(\alpha+sp_{iv}\) with \(s\simeq-18\). If malaria incidence is plotted as a function of this linear relation, we uncover a transition from disease persistence to elimination at \(\alpha-18p_{iv}\sim-1\). Our results suggests that an increase in malaria transmission due to heterogeneity can be overcome when the population fraction under ivermectin treatment is such that \(p_{iv}\gtrsim(\alpha+1)/18\).  
It is important to stress that our model predicts a higher impact of ivermectin in disease transmission when compared to previous empirical results [38]. Our agent-based model is a simplified version of real events concerning the consequences of iterating mosquito bites in human individuals, in a closed and well controlled simulation environment. However, despite this limitation as a tool for exploring hypothetical scenarios, the ivermectin algorithm in the model simulation presents  
Figure 3: (**a**) Malaria incidence as a function of the heterogeneous affinity \(\alpha\) and the ivermectin probability \(p_{iv}\) showing that isolines of malaria spreading are approximately a linear combination of both variables. (**b**) Malaria incidence as a function of the linear combination \(\alpha+sp_{iv}\) with \(s=-18\), showing a transition from disease persistence to elimination. In all cases, the human-to-mosquito transmission efficiency is held constant at \(w_{h}=0.5\).  
realistic results at a more qualitative level, in compliance with the expected ivermectin-induced mosquito mortality.  
1.2 The Role of Transmission Efficiency Combined with Ivvermectin Treatment in the Heterogeneous Scenario  
In the present model, human-to-mosquito transmission efficiency \(w_{h}\) is related to the time of gametocytemia duration and reveals a strong correlation with malaria incidence in high heterogeneity scenarios. However, this correlation is consistently suppressed with higher rates of ivermectin prevention, as shown in Figure 4a.  
In this case, as shown in Figure 4b, isolines of constant malaria incidence fit approximately a linear relation \(w_{h}+sp_{iv}\) with \(s\simeq-4.2\), and malaria incidence is approximately proportional to this linear combination.