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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