# Mathematical Model for Malaria Connects Regional Temperature to Mosquito and Parasite Traits

Malaria is a deadly infectious disease that is caused by parasites, which are transmitted indirectly to humans via mosquitos. “When I think of malaria, I always think of it as having these three populations that are interacting,” Miranda Teboh-Ewungkem of Lehigh University said. “You look at three interacting populations, each with their own lifespan and lifecycle. They’re all living populations trying to survive.” During a minisymposium presentation at the 2022 SIAM Conference on Mathematics of Planet Earth, which is taking place this week concurrently with the 2022 SIAM Annual Meeting, Teboh-Ewungkem described a mathematical model that relates parameters for parasites and mosquitos to temperature and explored the effects on malaria transmission.

In the lifecycle of the parasites that causes malaria, an infectious mosquito bites a human and deposits the parasite into their bloodstream. The parasite then divides and infects the red blood cells, eventually producing gametocytes that are transmissible to other mosquitos when they bite the human. The parasite then goes through different developmental stages in the mosquito before it is ready to be deposited in a new human’s bloodstream. Temperature affects a number of areas in this process: the mosquito’s biting rate, lifespan, and infectiousness through affecting the parasite’s development in the mosquito.

Figure 1. Temperature-dependent mosquito and parasite life-history traits. Black dots are empirical data from [1, 2] and red curves are functional fits. The mortality rate is from the inverse of data from [1] on mosquito lifespan. The mosquito growth rate resulted from multiplying the eggs per female per day production rate by the egg-to-adult survivorship. Top row, left to right: birthing rate, parasite development rate, and mortality rate versus temperature. Bottom row, left to right: eggs per female per day, egg-to-adult survivorship, and mosquito growth rate versus temperature. Figure courtesy of Miranda Teboh-Ewungkem.
It takes about ten days for the parasite to form in the mosquito, so by the time that the mosquito becomes infectious, it may be near the end of its life. However, the arrival time to infectiousness is shorter at higher temperatures — meaning that the mosquito has more opportunities to infect humans when the temperature is hotter. However, temperature also affects the larval stage of the mosquito, so the relationship with heat is not entirely simple.

Teboh-Ewungkem explained the compartmental model that she used to represent susceptible, infectious, and recovered humans. “We used a two-strain model that includes drug-sensitive and drug-resistant strains of malaria,” she said. “We actually applied it by fitting reported data to the model.” Humans in various model compartments interact with mosquitos, which is how the infection passes to and from the mosquito. The mosquito disease framework then comprises a smaller compartmental model.

Four particular model parameters were very important when accounting for temperature: mosquito biting rate, parasite development rate, mosquito death rate, and mosquito growth rate (see Figure 1). As the parasite development rates increases with increasing temperature, the timeframe needed for the mosquito to become infectious is shorter. The model needed to capture the fact that a mosquito will not infect a human that it bites before the parasite has had time to develop. Biting rate also goes up as temperature increase. Furthermore, the growth rate of the mosquito—depending on the number of eggs at the breeding site and how many survive to adulthood—is also affected by increasing temperature.

Figure 2. The temperature-dependent to time-dependent parameter traits for Blantyre (top two rows) and Chikwakwa (bottom two rows). Figure courtesy of Miranda Teboh-Ewungkem.
To investigate the seasonality of malaria with this model, Teboh-Ewungkem and her collaborators chose two regions in southern Malawi: Chikwakwa, which has high malaria transmission, and Blantyre, which has low transmission. The two regions have drastic differences in the temperature profiles, and Chikwakwa is overall much hotter.

“What we did then is say ‘Well, let’s think about it this way; we’re going to use a spline methodology, and before we do that we look at temperature-dependent parameters,’” Teboh-Ewungkem said. The collaborators used data collected in the lab on temperature-dependent traits in mosquitos and parasites—like biting rate, parasite development rate, mosquito mortality, eggs laid per day, and egg-to-adult survivorship—as a proxy for the model leading into the spline methodology.

“So, what does a spline methodology do?” Teboh-Ewungkem said. “It maps the temperature. I take a data point—for example, parasite development—and I know what the temperature profile is. I map the timeframe of that temperature profile to a time, so we are taking a temperature-dependent parameter and converting it to a time-dependent parameter which matches the profile of the temperature-time in a given locality.”

The collaborators implemented this process for Chikwakwa, mapping the temperature-time dependent data via the spline methodology to create a profile in Matlab (see Figure 2). They found the biting rate in January to be about 0.3 per day, while the rate is 0.2 in June and July (when the temperature in Chikwakwa is much lower). In addition, they found that more mosquitos die at higher temperatures. They then applied the same process in Blantyre, which produced very different profiles than Chikwakwa. “What was unique about this was you now can see the effect of temperature on parasites,” Teboh-Ewungkem said. “We can actually capture how each of these parameters are going to affect our model.”’

Teboh-Ewungkem was particularly interested in the monthly entomological inoculation rate, i.e., the number of infectious mosquito bites that a human receives each month. She wanted to know what happened to this parameter if the temperature increased or decreased by one or two degrees (see Figure 3). For comparison, the real-world data indicates that the peak malaria period in Chikwakwa is between January and April. If the model allows the temperature to increase by one degree, the height of the peaks increase and the malaria transmission in some periods that usually have lower malaria activity also start to increase. If the temperature increases by two degrees, malaria starts to cease to have a seasonal pattern and rather becomes present all year. This could be a serious impact of shifting climatic patterns. On the other hand, when decreasing the temperature by one or two degrees, the entomological inoculation rate for malaria also decreases.

Figure 3. Monthly entomological inoculation rate over time for Chikwakwa (left column) and Blantyre (right column). Rows from top to bottom: two degrees cooler than normal, one degree cooler, baseline temperature, one degree warmer, and two degrees warmer. Figure courtesy of Miranda Teboh-Ewungkem.
The researchers also changed the temperature for Blantyre, which usually only has a peak in malaria around January; the caseload is otherwise low. Increasing the temperature increased malaria overall, extending the peak season beyond just January. Another interesting aspect was that when looking at the fraction of infections that occur over a year, the peak period does not occur during the hottest period, at which time both the speed at which parasites develop and the mosquito mortality are increased.

Teboh-Ewungkem also looked at the portion of infections that are sensitive versus resistant strains; most of the infections were still sensitive. In Chikwakwa, simulations did predict that the region of coexistence between the sensitive and resistant strain would grow larger over time.

“Overall, at baseline monthly temperatures for Chikwakwa and baseline epidemiological parameters suggest a non-monotonic relationship between transmission potential and temperature shift,” she said. Higher temperatures often but do not always mean that malaria-carrying bites will increase — while the heat does decrease the time for parasites to develop, it also increases mosquito mortality. The model must account for the interplay between many different factors. “The key point of this is that we are actually using data that is driven by temperature to embed in the model and see how shifts in temperature are going to affect malaria infection,” Teboh-Ewungkem said.

Acknowledgements: Miranda Teboh-Ewungkem’s coauthors on the minisymposium presentation are Olivia Prosper (University of Tennessee), Katharine Gurski (Howard University), Angela Peace (Texas Tech University), Zhilan Feng (Purdue University), Margaret Grogan (United States Military Academy at West Point), and Carrie Manore (Los Alamos National Laboratory).

References
[1] Miazgowicz, K.L., Shocket, M.S., Ryan, S.J., Villena, O.C., Hall, R.J., Owen, J., … Murdock, C.C. (2020). Age influences the thermal suitability of Plasmodium falciparum transmission in the Asian malaria vector Anopheles stephensi. Proc. R. Soc. B, 287(1931), 20201093.
[2] Mordecai, E.A., Paaijmans, K.P., Johnson, L.R., Balzer, C., Ben-Horin, T., de Moor, E., … Lafferty, K.D. (2013). Optimal temperature for malaria transmission is dramatically lower than previously predicted. Ecol. Lett., 16(1), 22-30.

 Jillian Kunze is the associate editor of SIAM News.