# B Model of Malaria

Due to the two-host system, malaria is modeled using two dierential equations to describe the dynamics of infected humans and infected mosquitoes. These were rst modeled by R. Ross (1910) and then expanded by Macdonald et al. (1957). The model used in this paper is a Ross-Macdonald type model based on models used in D. L. Smith and McKenzie (2004), Cosner et al. (2009) and Torres-Sorando and Rodriguez (1997). There are two state variables and the model is usually expressed in continuous time, although here, I will present it in discrete time. The state variables are _yit_ , the fraction of mosquitoes infected in location i at time t and _xit_ , the fraction of humans infected in location i at time t.

In the model extension here that includes the impact of population movement, cases can either be generated locally in location i or imported from any other location j into i, _Ii_ . As modeled by Torres-Sorando and Rodriguez (1997), imported cases enter the model linearly. Local cases in location i are generated based on the susceptible population and several biological parameters. The susceptible population, _Sit−w_ , is the fraction of the population that is susceptible to malaria at time _t − w_ , where _w_ is the total parasite incubation period. The other parameters are the transmission eciencies from infected mosquitoes to humans and humans to mosquitoes, _bit−w_ and _cit_ , the number of bites on humans per mosquito, _ait−w_ and the ratio of mosquitoes to humans, _mit−w_ . The change in the number of infected humans and mosquitoes is described by: where _μit_ is the mortality rate of mosquitoes and _τit_ is the incubation period from the time a mosquito becomes infected until it is infectious. The parameter _ri_ is the recovery rate of humans. I make four assumptions arising from the epidemiology literature based on modeling the system in a low malaria setting:

1. The total parasite incubation time is one month based on the malaria cycle.

2. The mosquito population is at the steady state since mosquito populations have a relatively rapid turnover<sup>30</sup> 3. All malaria cases in month t are treated immediately and recover in month t.<sup>31</sup> 4. Based on D. L. Smith and McKenzie (2004), when the proportion of infected humans is small, the number of infectious bites received per day by a human (known as the entomological inoculation rate, taking the form _EIRit_ =<sup>_mit_</sup> _μ_<sup>_a_</sup> _<u>it</u>_<sup>2</sup> _it_<sup>_c_</sup> +<sup>_it_</sup> _a_<sup>_e_</sup> _it_<sup>_−_</sup> _c_<sup>_μitτit_</sup> _itxit_<sup>_xit_</sup> ) can be approximated by _citCitxit_ , where _Cit_ is the expected number of humans infected per infected human per day, assuming perfect transmission eciency ( _bit_ = _cit_ = 1 ), known as the vectorial capacity. The assumption applies because the analysis is conducted

> 30According to the CDC, adult female mosquitoes, which spread malaria, do not live more than 1-2 weeks in nature (Center for Disease Control 2015).

> 31The incidence data in the analysis is based on diagnosed cases, which are provided with free antimalarial treatment upon diagnosis. The literature shows that within a few days of treatment the majority of parasites are eliminated (Nosten and Nicholas J White 2007, N. White 1997).