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Paul
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Joined: July 20th, 2001, 3:28 pm

Models for Covid-19 - analytics

March 26th, 2020, 9:18 pm

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.
 
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Paul
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Re: Models for Covid-19 - analytics

March 26th, 2020, 9:49 pm

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.
 
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Paul
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Re: Models for Covid-19 - analytics

March 26th, 2020, 11:15 pm

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.
 
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Paul
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Re: Models for Covid-19 - analytics

March 28th, 2020, 4:30 pm

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.)
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