SERVING THE QUANTITATIVE FINANCE COMMUNITY

Paul
Topic Author
Posts: 10401
Joined: July 20th, 2001, 3:28 pm

### Models for Covid-19 - analytics

Continuing on from the numerical analysis thread...

Can solve for $I_i$:

$I_i(t)=e^{-\lambda t}\left( \lambda \int_0^t e^{\lambda \tau}S_i(\tau)d\tau-e^{\lambda t}S_i(t)+S_i(0)+I_i(0) \right).$

And then

$\frac{dS_i}{dt}=\left(\alpha-\frac{\beta}{N} \right)S_i\left( S_i- \lambda \int_0^t e^{\lambda (\tau-t)}S_i(\tau)d\tau -e^{-\lambda t}(S_i(0)+I_i(0))\right)\\ \qquad + \frac{\beta}{N}S_i\left( S- \lambda \int_0^t e^{\lambda (\tau-t)}S(\tau)d\tau -e^{-\lambda t}(S(0)+I(0))\right).$

Observe where there are and aren't subscripts.

When $\beta=0$ there is total lockdown and you can integrate each $S_i$ a bit. (*) Then you need to sum over all $i$ and you'll get $I$ which feeds into the $H$ and $D$ equations. Messy. So that's solve first then sum over $i$.

When $\alpha=\frac{\beta}{N}$ there is no lockdown. You can sum the $S_i$ equations over all $i$ and get the same result as in (*) but for the sum instead of individual households. So that's sum over $i$ then solve. (The other way around from above.)

For other cases you can't solve/sum in either order because of the $S_i^2$ term.

Paul
Topic Author
Posts: 10401
Joined: July 20th, 2001, 3:28 pm

### Re: Models for Covid-19 - analytics

Or in two lines if I was able to concentrate(!):

$\left( \frac{S_i'}{S_i} \right)'+\lambda \frac{S_i'}{S_i}=\left(\alpha-\frac{\beta}{N} \right)S_i'+\frac{\beta}{N}S'.$

So

$\frac{S_i'}{S_i} +\lambda \ln(S_i)=\left(\alpha-\frac{\beta}{N} \right)S_i+\frac{\beta}{N}S +c_i,$

where

$c_i=\lambda \ln(S_i(0))-\left(\alpha-\frac{\beta}{N}\right) \left(S_i(0)+I_i(0) \right)-\frac{\beta}{N}\left(S(0)+I(0) \right).$

From this

$(\ln(S_i/S_i(0)))'+\lambda \ln(S_i/S_i(0))=\\ \qquad\left(\alpha-\frac{\beta}{N} \right)S_i+\frac{\beta}{N}S-\left(\alpha-\frac{\beta}{N}\right) \left(S_i(0)+I_i(0) \right)-\frac{\beta}{N}\left(S(0)+I(0) \right).$

Which can (almost) be solved for those two extreme cases.

Paul
Topic Author
Posts: 10401
Joined: July 20th, 2001, 3:28 pm

### Re: Models for Covid-19 - analytics

PROBABLY ERRORS IN ALL THIS

Simple case: $n$ households infected the same $S^{(1)}$ and the rest uninfected $S^{(0)}$.

$(\ln(S^{(0)}))'+\lambda \ln(S^{(0)})=\\ \qquad\left(\alpha-\frac{\beta}{N} \right)S^{(0)}+\frac{\beta}{N}\left(nS^{(1)}+(N-n)S^{(0)} \right)-\alpha.$

$(\ln(S^{(1)}/(1-I^*)))'+\lambda \ln(S^{(1)}/(1-I^*))=\\ \qquad\left(\alpha-\frac{\beta}{N} \right)S^{(1)}+\frac{\beta}{N}\left(nS^{(1)}+(N-n)S^{(0)} \right)-\alpha.$

Rescaling $S^{(1)}=(1-I^*)\bar{S}^{(1)}$ so that both (new) $S$s start at 1:

$(\ln(S^{(0)}))'+\lambda \ln(S^{(0)})=\\ \qquad\left(\alpha-\frac{\beta}{N} \right)S^{(0)}+\frac{\beta}{N}\left(n(1-I^*)\bar{S}^{(1)}+(N-n)S^{(0)} \right)-\alpha.$

$(\ln(\bar{S}^{(1)}))'+\lambda \ln(\bar{S}^{(1)})=\\ \qquad (1-I^*)\left(\alpha-\frac{\beta}{N} \right)\bar{S}^{(1)}+\frac{\beta}{N}\left(n(1-I^*)\bar{S}^{(1)}+(N-n)S^{(0)} \right)-\alpha.$

Two coupled ODEs.

Paul
Topic Author
Posts: 10401
Joined: July 20th, 2001, 3:28 pm

### Re: Models for Covid-19 - analytics

Random thoughts:

Is it better to have simpler model for in-household infection? Integers? Or assume everyone infected? So only two types of household? Depends on timescales (in-house and outhouse).

In our household only one person goes out to shop. Suggests to me that there is an optimum household size. (Will depend on how many currently infected.)