SI model with no demographics¶

We will begin with perhaps the simplest possible model of an infectious disease - the SI model without demography. This model has two classes of individuals, the infective and the susceptible, and 1 parameter, $\beta$, describing the per-timestep, per-infective person rate at which susceptibles transition to infectives.

$$ \dot{S} = -\frac{\beta*S*I}{N} $$

$$ \dot{I} = \frac{\beta*S*I}{N} $$

Noting that $S = N-I$, we can substitute $S$ in the second equation to get a standard form logistic equation

$$ \dot{I} = \beta \frac{(I-N)I}{N} = \beta I (1 - \frac{I}{N}) $$

With solution $$ \frac{N}{1+(\frac{N}{I_0}-1)e^{-\beta t}} $$

To run the model in a discrete, constant time-step solver, the differential equation above is converted into a stochastic finite difference equation

$$ \Delta I = Bin(S_t, 1-exp^{-\beta \Delta t \frac{I}{N}}) $$

$$ S_{t+1} = S_t - \Delta I $$

$$ I_{t+1} = I_t + \Delta I $$

This notebook tests the implementation and behavior of the model as follows:

Construct the model¶

In the first few cells, we import the laser.generic.SI model. Then we construct a single-patch LASER model with two components: the Susceptible and Infected states, and the Transmission step in which Infecteds can infect Susceptibles. Finally, we initialize with a single infection and run.

Note that LASER-generic has separate importable models for different generic model types (SI, SIS, SIR, SEIR, etc.). Model code can be imported or copied over from one "sub-package" to another as desired, but maintaining this separation enables model-specific performance optimizations and cleaner code than requiring the components to "know" the global model configuration and change internal behvaior based on that (for example, whether agents in the Infected state need to cound down a timer or not; whether Transmission should send newly infected agents to the Infectious state as in the SI, SIS, SIR, or to an Exposed state as in an SEIR).

Sanity check¶

The first test ensures certain basic constraints are being obeyed by the model. We confirm that at each timestep, $S_t=N_t-I_t$. We also confirm that $I_t = I_0 + \sum_{t'=0}^t \Delta I_{t'}$

Scientific test¶

Finally, we come to the scientific test. We select a few values of $\beta$, run the model, fit the outputs to the logistic equation, and compare the fitted value of $\beta$ to the known value. Because we are approximating continuously compounding growth, in the logistic equation, with a discrete time-step approximation, we expect the fitted values of $\beta$ to be biased slightly downward - that is, the modeled trajectory is slightly slower than the continuous-time counterpart. This error grows as $\beta$ gets larger; the test fails if any of the fitted $\beta$ values are more than 5% away from the known values.

Constructing the model¶

A note that order of the model components matters, because of how recording of relevant statistics into outputs occurs. If transmission comes before susceptibility, then we have N=S+I+ΔI, because we record I, record ΔI, do transmission, then record S after the transmission process occurs. With susceptibility first, we record S, then record I, then do transmission (and record ΔI), and so we have N=S+I.

First set of sanity checks¶

Introducing stochasticity into a model can have more effects on an individual model realization than just inducing noise around a mean trajectory - this simple model is a great example. In the early stages of the outbreak, the infection count is low, and the timing of those first few infections dictates when the logistic growth really takes off. This appears as a random time-shift in the entire outbreak trajectory.

To illustrate this phenomenon, the below plot shows the model output, the expected logistic growth curve, and the expected logistic growth curve fit to the model with one free parameter - an offset $t_0$. The resulting plot should show good concordance between the model output and the expected logistic equation with the known model inputs $\beta$ and population.

The goodness of this fit will be turned into a strict pass/fail test down the line.

Scientific testing¶

Finally, we run the model for a range of β parameters, we freely fit the model output to the logistic equation, and we compare the known input parameters against the parameters fitted from output.

Because we are approximating continuously compounding growth by discrete-time compounding growth, we should expect the fitted β to consistently be slightly underestimated relative to the true β , with the relative difference growing as β gets larger.

In the future, we could probably compute the expected error from this approximation. But for now, to make this a pass-fail test, we will raise a flag if the fitted β is more than 5% different than the known β.