The simplest model of malaria assumes that the mosquito population is at equilibrium and (only) models the infected humans $I$ with the following equation:

$$\frac{dI}{dt} = \frac {𝛼𝛽𝐼}{𝛼𝐼+π‘π‘Ÿ}(π‘βˆ’πΌ)βˆ’πœ‡πΌ$$

where π‘Ÿ is the natural death rate of mosquitoes, $πœ‡$ is the death rate of humans, $𝛽$ is the transmission rate from infected mosquitoes to susceptible humans, and $𝛼$ is the transmission rate from humans to mosquitoes.

Find the equilibria of this model and determine stability conditions for the disease free equilibria to be stable or unstable.

So I know that for this DE to be at equilibrium $\frac {dI}{dt}=0$, and the only way I can come up with for this to be at equilibrium is when $I=0$ ie. the disease free equilibria. I am having difficulty finding any other equilibria. Also, how do I go about determining the stability conditions for the disease free equilibrium ($I=0$, I think) to see whether it is stable or unstable.

  • $\begingroup$ It seems to me that you can find another equilibrium by multiplying both sides of the equilibrium equation by $\alpha I + N r$ and solving a quadratic equation. $\endgroup$ – Ian Oct 6 '16 at 23:48
  • $\begingroup$ Which corresponds to the state where the number of newly-infected humans is equal to the number of infected humans dying. $\endgroup$ – ConMan Oct 7 '16 at 0:25
  • $\begingroup$ when i solve for the quadratic equation here i get a total mess.. any tips? $\endgroup$ – user371517 Oct 7 '16 at 0:46
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    $\begingroup$ I mean, the actual characterization of that equilibrium will be a pretty nasty mess, given the complexity of your growth term. I don't think that's going anywhere. $\endgroup$ – Ian Oct 7 '16 at 0:54
  • $\begingroup$ Actually the determination of the nature of each equilibrium is completely standard, as is often the case for 1D models, since only the sign of the RHS matters. $\endgroup$ – Did Oct 10 '16 at 20:12

You need $$ \frac {\alpha\beta I}{\alpha I+Nr}(N-I)-\mu I = 0. $$ If $I\ne 0$ then you can divide both sides by $I$: $$ \frac {\alpha\beta}{\alpha I+Nr}(N-I)-\mu = 0. $$ Can you solve that for $I$?

  • $\begingroup$ yes so that would be the other equilibria then (solved in terms of I)? $\endgroup$ – user371517 Oct 7 '16 at 1:15
  • $\begingroup$ Yes. But the singular is "equilibrium". $\qquad$ $\endgroup$ – Michael Hardy Oct 7 '16 at 1:24

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