# Stability

## External perturbations

Suppose an intervention measure has successfully stopped the spread of an epidemic, and for the moment there are no infected people left. Part of the population is immune due to having contracted the disease, another part is not. At this point, the restrictions due to the intervention measure can be lifted.

But what happens to this situation when there is an external perturbation? For instance, someone gets infected on vacation abroad an returns, carrying the infection. Mathematically, this is asking for the stability of the current situation (distribution of immune people as well as active intervention measures) - if the situation is not stable, the epidemic propagates again across the whole grid after a perturbation, if the situation is stable the perturbation can cause a small localized outbreak, but is no longer able to affect the whole grid.

To a degree, this question is related to prior immunity - which we have covered already - but we have already stated that the spatial distribution of immune people due to the disease forms a characteristic honeycomb structure which is markedly different from a purely random distribution. So while the immune_fraction keyword allows to test the stability of a completely random spatial distribution, we're now interested in testing the stability of a more complicated situation.

## Seedings

Create the following config file as the baseline epidemic:

 simulation num_timesteps 1800 snapshot_interval 200 filename_base stability random_seed 101 disease recovery_time 14 p_transmission 0.14 population grid_size 1000 immune_fraction 0.0 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 seedings num_seedings 1 seeding time 800 number 50 transmission 0.14 end

Note the seedings and seeding keywords - they allow to define a perturbation in terms of a number of seeds (here 50) which are inserted into the grid at a specified time (here timestep 800) with the transmission fraction of the defined disease.

The scenario starts the epidemic and lets it spread, then inserts 50 new infections onto random grid positions at timestep 800.

Running this basic scenario infects about 75% of the population before timestep 600, at which point the propagation stops. Inserting the new seeds later causes no visible new outbreak.

Now, let's try to stop the spread of the epidemic early by a relatively hard intervention measure. Insert the block

 measures num_measures 1 measure start 200 duration 100 transmission 0.03

into the scenario and re-run. The epidemic now stops spreading around timestep 200, but produces a violent secondary outbreak around timestep 900 when the seeds grow into substantial outbreaks. Apparently it is highly unstable.

Now, let's vary the intervention time from 200 to 400 - you should see something like this:

 Stability as a function of intervention time.

As the initial outbreak is allowed to grow more, the secondary outbreak gets progressively milder. This is expected - if the first outbreak causes a large fraction of immunity in the population, there is less room for a seconary outbreak. What is perhaps less expected is that the sum of first and second outbreak is usually smaller when an intervention is used than without intervention, and the smallest value is reached when the intervention is done at about half of the number of infected that would be reached without (you can check by varying the random seed and re-running the scenarios that this is no mere quirk).

## Conclusions

If you try a bit with different mobility parameters and containment strategies, the pattern that emerges is as follows: It is possible to create (relatively) stable distributions of immune people in the grid population due to intervention measures in which in the end less infections occur than without the measures. The degree to which this is possible depends on the randomness of the distributions - the more random the initial spread is and the more flamefront-like the secondary spread is, the less infections are counted in the end. Thus, the prior immunity scenario we've looked earlier was a best-case scenario.

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