# Varying-coefficient SIRD model

In my post “A Tale of Two Cities” I used a varying-coefficient SIRD model for curve fitting. This is a variant of the standard SIRD model. Specifically, in a population of size $$N$$, at a given time $$t$$ there are $$S(t)$$ susceptible individuals (not infected yet), $$I(t)$$ infectious individuals (individuals who are infected and actively infecting others), $$R(t)$$ removed individuals (who were infected but are no longer infecting others, because they have either recovered or isolated themselves), and $$D(t)$$ dead individuals. So $$S(t)+I(t)+R(t)+D(t)=N$$ at all times. The observable quantities here are $$S(t)$$ and $$D(t)$$, which we can get from confirmed positive cases ($$N-S(t)$$) and deaths. The other two, $$I(t)$$ and $$R(t)$$, are not directly observable. Although some cities and countries report the number of recovered individuals, this is not the same as $$R(t)$$, because $$R(t)$$ includes, in addition to the recovered, those who are still sick but in isolation.

The varying-coefficient SIRD model is given by the system of differential equations

\begin{align} S'(t) &= - \beta(t) I(t) \frac{S(t)}{N}, \\ I'(t) &= \beta(t) I(t) \frac{S(t)}{N} - \gamma I(t) - \mu I(t), \\ R'(t) &= \gamma I(t), \\ D'(t) &= \mu I(t). \\ \end{align}

Here we use a $$\beta(t)$$ that is a function of time rather than a constant. The reason is that this parameter $$\beta$$, the number of infecting contacts per individual, is the one that quarantine-like measures are intended to modify, so it is expected to vary with time. The other two parameters, the removal rate $$\gamma$$ and the death rate $$\mu$$, can be assumed constant, in principle.

The function $$\beta(t)$$ itself can be modeled, for example, as a spline function. I used cubic splines in my analysis. This way the model calibration problem becomes a common multivariate estimation problem, and the parameters can be estimated by least squares from the data.

Since $$\beta(t)$$ is a function of time, the basic reproduction number $$R_0(t)$$ will also be a function of time,

$R_0(t) = \frac{\beta(t)}{\gamma + \mu}.$

# When does the curve turn around?

From the second equation above we see that the inflection point $$I'(t)=0$$ occurs when $R_0(t)\frac{S(t)}{N} - 1 = 0,$ so $$I(t)$$ decreases when $R_0(t)\frac{S(t)}{N} < 1.$

This occurs either if $$R_0(t)<1$$, regardless of the proportion of susceptibles $$S(t)/N$$, or if the proportion of susceptibles $$S(t)/N$$ satisfies $\frac{S(t)}{N} < \frac{1}{R_0(t)}.$ Since $$S(t)$$ decreases in time, epidemics always come to end, sooner of later: some because $$R_0(t)<1$$ and some because the herd-immunity threshold $$1/R_0(t)$$ is attained.

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