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The simplest malaria model is as follows: $$\frac{dI}{dt} = \frac{\alpha \beta I}{\alpha I + r} (1-I) - \mu I$$

where $r$ is the natural death rate of mosquitoes, $\mu$ is the death rate of humans, $\beta$ is the transmission rate from infected mosquitoes to susceptible humans, and $\alpha$ is the transmission rate from humans to mosquitoes. However, for $I \ge 1$, $dI/dt$ is negative. Wouldn't that imply that the infected class is always shrinking from the start? How does this make sense?

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    $\begingroup$ What is I? Overall infections? $\endgroup$ – GrumpyMammoth Nov 20 '17 at 5:54
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    $\begingroup$ Yes, $dI/dt$ is the rate of change of the infected population. $\endgroup$ – Derek Adams Nov 20 '17 at 12:43
  • $\begingroup$ Where did the model come from? I'm thinking hard trying to figure this out, maybe a bit more background could help. $\endgroup$ – GrumpyMammoth Nov 20 '17 at 13:01
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    $\begingroup$ Martcheva, Introduction to Mathematical Epidemiology. $\endgroup$ – Derek Adams Nov 20 '17 at 13:33
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I am afraid you may have to read your textbook more closely. The question on page 29 states that this equation models the proportion of infected humans. If $I$ $≥1$ that would mean that there are more people are infected than there are people in total - e.g. 3/2 of people are infected - this clearly doesn't make sense.

Thus $1≥I$ when using this model.

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    $\begingroup$ Ah, thank you very much for clarifying. You've spared me much frustration! $\endgroup$ – Derek Adams Nov 20 '17 at 14:15

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