March 26th, 2020, 5:47 pm

**This is the full lockdown+hospital model. The version to work on. Only change from previous version is putting in the N parameter to help with scaling.**

Independent variable:

[$]t[$] is time.

Dependent variables:

[$]S_i(t)[$] = susceptible in each household, at home

[$]I_i(t)[$] = infected in each household, at home

[$]R_i(t)[$] = recovered (alive) in each household, at home

[$]H_i(t)[$] = in hospital from each household

[$]D_i(t)[$] = dead from each household

**Only the first two sets of these are coupled. So it is not as bad as it looks. And they differ across [$]i[$] only in initial conditions. The equations and parameters do not vary with ****[$]i[$].**

[$]i[$] is a household index, [$]i=1,\ldots,N[$], so there are [$]N[$] households. This means that each household has its own five quantities ([$]S,I,R,H, D[$]).

There are five ODEs:

[$]\frac{dS_i}{dt}=-\left(\alpha -\frac{\beta}{N}\right) S_i I_i-\frac{\beta}{N} S_i I\qquad[$] (1)

[$]\frac{dI_i}{dt}=\left(\alpha -\frac{\beta}{N}\right) S_i I_i+\frac{\beta}{N} S_i I-\gamma I_i-\delta I_i\qquad[$] (2)

[$]\frac{dR_i}{dt}=\delta I_i+\epsilon H_i\qquad[$] (3)

[$]\frac{dH_i}{dt}=\gamma I_i-\epsilon H_i -\phi(H) H_i\qquad[$] (4)

[$]\frac{dD_i}{dt}=\phi(H) H_i\qquad[$] (5)

Here

[$]I(t)=\sum_{j=1}^NI_j(t)[$], meaning the total population of infected across all households,

and

[$]H(t)=\sum_{j=1}^NH_j(t)[$], meaning the total number of people in hospital across all households.

Whenever the subscript on a dependent variable is missing it means summed over all [$]i[$].

**Only two sets of these ODEs are coupled. **

These are the meanings of the parameters:

[$]\alpha[$] is the usual SIR parameter for susceptible becoming infected, within household

[$]\beta[$] is the usual SIR parameter for susceptible becoming infected, outside household and therefore much smaller than [$]\alpha[$] during lockdown

[$]\gamma[$] represents going into hospital.

[$]\delta[$] for recovered without hospitalization

[$]\epsilon[$] for returning from hospital.

There is a function here, [$]\phi[$], for deaths in hospital, that we have to specify. It is a function of total number of people in hospital.

[$]\phi\left( \sum H_i\right)=\phi(H)[$]

Solve coupled equations (1) and (2). They feed into (4) and (3) then (5).

The parameter values I used for my second two plots were

[$]\alpha=2[$]

[$]\beta/N=2[$] for the first plot, 0.2 for the second

[$]\gamma = 1.5[$]

[$]\delta =1[$]

[$]\epsilon = 1[$]

For the [$]\phi[$] function I had

[$]\phi(H) = 0 \;\mbox{for}\;H<1, 2\;\mbox{for}\;H>1[$]

I had ten households, so [$]N=10[$]

For initial conditions I had [$]R_i(0)=H_i(0)=D_i(0)=0[$].

And I had uniformly random [$]I_i(0)[$] between zero and [$]I^*[$]. Then [$]S_i(0)+I_i(0)=1[$]. (I don't know whether changing household size matters.)

In my examples I had [$]I^*=0.2[$].

Notes:

You don't need the third equation, for [$]R_i[$]. It's there because it's in the SIR model, and in case there is the possibility of either reinfection or the recovered passing on the disease. That would require tweaking the model.

Total number of deaths is found by solving

[$]\frac{dD}{dt}=\phi(H) H[$]

**Please play around with parameter values and the [$]\phi(H)[$] function. My parameters were not carefully thought out, just used to see it I could get any results at all.**