# Prior immunity

## When does it stop?

Evidently, the epidemic never seems to affect all if the simulated population (which would be a million)) - most scenarios we've seen so far seem to stop at around 2/3 of that value. Can we understand that?

In the exponential picture at least, the epidemic continues to propagate as long as every infected person spreads to the disease to at least one other person during a cycle. So if the original R0 is 3.3, that means that the disease stops spreading when only a fraction of 1/3.3 (or 30%) of the population can carry the disease, i.e. after 70% have been affected and are immune. That more or less seems to play out in the simulation.

So - what about some prior immunity? Assume people are immune from a previous wave of the disease, or have been vaccinated - how much vaccination is needed to stop the disease? Based on the estimate above, we'd guess that we need to vaccinate some 60 to 70% of the population. Right?

Wrong.

## Prior immunity

Let's test this idea. Create the following config file:

 simulation num_timesteps 2000 snapshot_interval 200 filename_base vaccination_01 disease recovery_time 7 p_transmission 0.23 population grid_size 1000 immune_fraction 0.1 num_classes 3 class mobility 3 p_mobility 1.0 fraction 0.8 class mobility 5 p_mobility 1.0 fraction 0.19 class mobility 100 p_mobility 1.0 fraction 0.01 end

The immune_fraction keyword sets a random 10% of the population as a priori immune to the disease.

Run the scenario a few times for different value of the prior immunity, and you'll find something like this:

 Comparison of different prior immunity levels.

A mere 15% prior immunity are enough to pretty much stop the disease in its tracks. What is going on here? Has this to do with flamefront effects? Let's make the mobility high in all classes to get to the exponential scenario and re-run the simulation - you'll find something like this:

 Comparison of different prior immunity levels in the exponential case.

So now we need something like 45% of immune people to stop the spread - which means that partially there are spatial effects relevant, but partially there must be something else since 45% still is not 60%.

Well, the latter we can understand by arguing that what 15% immunity in the population changes is not the number of people a carrier infects but rather the probability - in 15% of the cases where a transmission would have happened otherwise, the new carrier turns out to be immune, so the transmission probability is reduced by 15%. But the reduced probability is under an exponential, which means that it does not reduce the number of newly infected carriers by 15% but by 23%, and by the same token a reduction of 40% in the probability translates to 75% under the exponential. Thus, we can well understand the exponential case.

To see how the flamefront case is different, let's look at some spatially resolved plots:

 Spatial distribution after 600 days with 10% prior immunity.

It turns out that the outcome of the simulation is a foam-like structure - there are large pockets to which the infection never spreads. So what happens in the case of prior immunity that in many places the advancing infection front is locally stalled when it happens to run into a random cluster of immune people - and many of these stalls combine to create protected pockets into which the epidemic never penetrates.

When the immunity level is increased even further, the stalling of the fronts is in fact such a frequent occurrance that the epidemic never spreads across the whole grid.

 Spatial distribution after 2000 days with 15% prior immunity.

## Consequences

Why is this dramatic effect not observed with acquired immunity and requires prior immunity? Because unlike prior immunity which is randomly distributed across the grid, acquired immunity is spatially correlated with the spreading front - in particular it is mostly behind the front where it is useless to stall the further advance. There's an analogy with counter-fires used to combat large forest fires - a layer of burnt land behind the front does not hold it in any way, but an area devoid of fuel in the direction of the front advance holds it very efficiently.

Thus, the actual effect of vaccination or any other prior immunity is, especially in the spatially detailed picture, much more pronounced than the naive estimate would suggest.

Continue with Containment measures.

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