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The simplest malaria model is given by $$\frac{dI}{dt} = \frac{\alpha \beta I}{\alpha I + r} (1-I) - \mu I$$ where $\mu$ is the death rate of humans, $\alpha$ is the transmission rate from humans to mosquitoes, $\beta$ is the transmission rate from infected mosquitoes to susceptible humans, and $r$ is the natural death rate of mosquitoes.

Assuming the units are given in $d^{-1}$, or per day, we can assume $r$ to be rather large, given that it isn't part of a larger product (and thus isn't a proportion). We can thus assume $r$ to be anywhere from a few dozen to a few hundred, depending on the size of the population. Recall that this is the amount of mosquitoes killed per day.

$\alpha$ is the transmission rate from humans to mosquitoes, and does appear to be a proportion, so $0 < \alpha < 1$. Thus, $\alpha$ is the probability a mosquito not carrying malaria bites a human $\textit{with}$ malaria. Assuming nothing more than 15% of a population of humans has malaria, and that 95% of a mosquito population do not carry malaria, this is $0.15*0.95 = \alpha = 0.143,$ or a 14% probability.

$\beta$ is also a proportion, and is the transmission rate from infected mosquitoes to susceptible humans. Rather, $\beta$ is the probability an infected mosquito will bite a human. Assuming a generous mosquito population (a few thousand at the least), we assume that about 5% carry malaria and (based on our previous assumptions) they have a 85% chance of biting a susceptible human (given that 15% are already infected). So $\beta = 0.05*0.85 = 0.0425$.

$\mu$ is given as the death rate of humans. Neglecting deaths by reasons other than malaria itself, we assume this value to be about $0.3\%$, given 627,000 deaths in 2012 out out of an estimated 207 million cases that year.

Is this analysis correct? If not, where can I find values for the above parameters? Are there any case studies where I can calculate the above values?

Thank you in advance.

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  • $\begingroup$ To use more accurate figures, currently only ~3% of the world's population has malaria, and, roughly 20% of mosquitos that are able to carry malaria actually live long enough to potentially [have and] spread the disease. $\endgroup$ – Charles Nov 21 '17 at 0:26

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