Model for spread of contagious disease.

[elitewarr]

Hello guys, I am quite unsure in how to start the modeling for population. If I define y as the number of infected persons and y' as the rate of spread is it correct? I see the number of contacts between infected and non infected persons to be equal to the number of infected persons. Otherwise I have no idea how to form the variables.

Thank you.

 

[economicsnerd]

If $p\in(0,1)$ is the infected proportion, it may be convenient to model things in terms of $x=\log\dfrac{p}{1-p}$, as the latter (which is in one-to-one correspondence with the former) can take any real value.

It sounds like what is being described is that $\dot p$ is proportional to $e^x$, so say $\dot p = \alpha e^x$ for some $\alpha>0$.

Then we can compute $\dot x e^x = \dfrac{d}{dt} e^x = \dfrac{(1-p)\dot p - p(-\dot p)}{p^2} = \dfrac{\dot p}{p^2}= \dfrac{\alpha e^x}{p^2}$, and so $\dot x = \dfrac{\alpha}{p^2}$. One can further verify that $p = \dfrac{1}{1+e^{-x}}$, and so $\dot x = \alpha(1+e^{-x})^2$.

From the above equation, it's easy to see that:
- $x$ is strictly increasing over time if $-\infty<x<\infty$, and therefore $p$ is strictly increasing if $0<p<1$.
- Therefore, $x$ must converge to something finite or infinite. It's easy to see that it can't converge to anything finite over time. Therefore $x\to\infty$. Thus $p\to 1$ as long as $0<p\leq 1$.

The only remaining case of $p$ starting at $0$ is easy to consider on its own.

 

[economicsnerd]

Oh whoops, it's simpler than that. Disregard.

Number of contacts between infected and uninfected should be proportional to $p(1-p)$ maybe?

 

[nsaspook]

http://arxiv.org/abs/1311.6376

Mathematical models of epidemic dynamics offer significant insight into predicting and controlling infectious diseases. The dynamics of a disease model generally follow a susceptible, infected, and recovered (SIR) model, with some standard modifications. In this paper, we extend the work of Munz et.al (2009) on the application of disease dynamics to the so-called "zombie apocalypse", and then apply the identical methods to influenza dynamics. Unlike Munz et.al (2009), we include data taken from specific depictions of zombies in popular culture films and apply Markov Chain Monte Carlo (MCMC) methods on improved dynamical representations of the system. To demonstrate the usefulness of this approach, beyond the entertaining example, we apply the identical methodology to Google Trend data on influenza to establish infection and recovery rates. Finally, we discuss the use of the methods to explore hypothetical intervention policies regarding disease outbreaks.

 

[elitewarr]

Sorry for the late reply. Thank you guys. But I think the simpler version should suffice according to the logistic equation.

And.. I don't rawly understand what you wrote economicsnerd haha.. My knowledge isn't that advanced yet..