# Rotavirus

We were tasked for the SISMID summer courses with doing a little analysis of a virus. Our group ended up with Rotavirus, a stomach bug. I was interested, as usual in time behavior, so I found a paper that had some time series data and started thinking about the dynamics.

1. The virus basically only infects 5 year olds. Immune protection is extremely successful so we might allow for 2 susceptible compartments having very different infectivity parameters.
2. The virus can live for 10-20 days in the environment, so there is a large reservoir in that sense as well.
3. The virus is very seasonal, which appears to both be due to faster viral degradation at higher temperatures, increased contact in colder climates when people are inside and closer to one another, and potentially even immune weakness due to vitamin D deficiency. See Celik et al. 2015 though I can’t understand why colder months have higher humidity.

Perhaps the system might be something like

$\dot{S}_k = \mu - \beta_k(t) S_k I - \alpha$
$\dot{S}_a = \alpha - \beta_a(t) S_a I$
$\dot{I}_k = \beta_k(t) S_k I - \gamma R_k$
$\dot{I}_a = \beta_a(t) S_a I - \gamma R_a$
$\dot{R}_i = \gamma (R_k+R_a)$
$\dot{V} = \pi I - \delta V$

given kids subscripted $k$ and adults subscripted $a$. We also have free virus $V$ that is produced proportional to the number of infectives and cleared with the rate $\delta$. Then the number of infecteds is $I=I_k+I_a+\phi V$, the sum of all infecteds and with a conversion rate between free virus. Then we would have that the infectivities obey $\beta_k(t) \gg \beta_a(t)$ but are both seasonally forced with some oscillating function of time. We have $\mu$ the birth rate, $\alpha$ the aging rate, $\gamma$, and the recovery rate which we assume to be the same for all.

I wanted to do some dynamical systems analysis, but actually the dynamics aren’t that interesting in the observed data… Maybe I’ll get back around to studying this system in more detail.