## 2 Methods, Model, and Analysis Tools
### Human-to-Mosquito Transmission and Drug Administration in a Heterogeneity Setting  
At least two mosquito bites are required to complete the human-mosquito-human malaria transmission cycle. However, it is the presence of gametocytes in the blood at the human-to-mosquito transmission stage that stands as the most critical step in the persistence of malaria transmission. Because human hosts can present a wide range of gametocytemia levels on different occasions and during a finite period of time, the effectiveness of human-to-mosquito transmission assumes a stochastic nature, conditioning some form of heterogeneity in the process. In the present model, human-to-mosquito transmission efficiency \(w_{h}\) is defined as the fraction between the number \(\tau_{g}\) of days with positive gametocytemia and the expected total duration \(\tau_{d}\) of the disease, as [24]:  
\[w_{h}=\frac{\tau_{g}}{\tau_{d}}\,. \tag{4}\]  
Here, we consider the typical value \(\tau_{d}=150\) days. The time period with potential gametocyte production is usually longer than the disease symptomatic phase, which is of a shorter duration. The assumed value of 150 days represents a reasonable approximation to the upper range of the expected average disease duration. Gametocytemia duration was defined as an essential model parameter, tunable through a dynamically equivalent action by the potential use of a gametocidal drug agent. In the present model, heterogeneity is analyzed at different levels of disease H-to-M transmission (from \(w_{h}=0.400\) to \(w_{h}=0.800\)). In the case of heterogeneous transmission, the combination of several anti-malarial strategies is crucial to achieve epidemic control [3]. The possibility of quarantine may be ineffective in malaria asymptomatic patients. Even gametocyte carriers may briefly show only asexual forms of the most dangerous parasite _Plasmodium falciparum_ (_Pf_) in their peripheral blood. With new laboratory assays for early disease diagnosis, diagnostic accuracy in migrant populations may improve. However, deciding which migrants should be treated with effective anti-malaria treatment at admittance is debatable. Ivermectin has proven effective against _Plasmodium falciparum_ infection, by selective killing of _Anopheles_ mosquitoes shortly after a successful bite, before the parasite is able to complete its sporogonic development [22; 34; 35; 36; 37; 38; 39], and apparently without any form of direct toxicity to the parasite [40]. This ivermectin selective killing action on biting mosquitoes may be extremely effective in disease prevention, by reducing the need for treatment with artemisinin derivatives, thus lowering the risk of artemisinin drug-resistance expansion [37]. Ivermectin prevention was introduced in the present model as an adjustable parameter \(\delta_{iv}\), according to a previously published definition [24]. The \(\delta_{iv}\) parameter results from the product of ivermectin administration probability \(p_{iv}\) (in the range between 0.0, with no ivermectin prevention, and 0.50, with ivermectin prevention in 50% of the human population) and the ivermectin-induced mosquito mortality constant parameter (\(g_{iv}=0.50\)):  
\[\delta_{iv}=p_{iv}\times g_{iv}\,. \tag{5}\]  
This mathematical implementation of the ivermectin effect is based on the knowledge that approximately half of all mosquitoes will die during the first four days after feeding on a human-host under ivermectin treatment [22]. As described below, we will validate such implementations in our model, looking into mosquito survival distribution curves and comparing them with [PERSON]'s original results. We also evaluate if there is any correlation between mosquito mortality as a result of ivermectin human treatment when compared to mosquito mortality occurring during the parasite incubation period inside the mosquito, according to what would be expected from theoretical predictions.