# Waves

## Types of waves

Often diseases strike in several discrete waves. This can happen for a number of reasons. The simplest one is that there is a factor in nature which modulates the transmission probability. This can for instance be the seasonal variation of the temperature, at cold temperatures the human immune system is more prone to contracting respiratory diseases.

Another reason may be the appearance of a more virulent disease strain that can re-kindle infections even if the original strain has exhausted its possibilities.

Yet another possibility is that the disease has been fought by containment measures and so infections are driven to low numbers, however the containment measures are then relaxed, which usually prompts a resurging wave. As we have seen in the tutorial on Stability, even driving the number of new infections to zero does not prevent such resurgence if external sources are present.

In the context of the simulation so far, waves can not continue forever as sooner or later the whole grid has been infected and is hence immune. To have year by year recurring waves (such as for influenza) requires either fading immunity against the original disease over time, or mutations against which a previous infection with a different strain does not confer full immunity any more.

To see and study wave behavior, it is most instructive to plot results as daily new infections rather than as cumulative number of infected people as we have previously done.

## Seasonal waves

The modulation of disease transmission probabilities over time can be implemented as measures. The highest transmission probability then (say for winter) gets assigned in the definition of the disease, and this is then lowered in the other seasons by measures.

To actually see seasonal waves on the grid when the time unit is assumed to be days requires some fine-tuning, as the first wave needs to be slow enough not to infect the whole grid within one season. Also, the unfavourable probability (say summer) needs to be high enough such that the disease does not yet die out. The following file can serve as a starting point for experiments.

 simulation num_timesteps 800 filename_base wave_seasonal random_seed 108 disease recovery_time 7 p_transmission 0.205 population grid_size 1000 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 10 number 300 transmission 0.205 measures num_measures 5 measure start 30 duration 58 transmission 0.20 measure start 91 duration 88 transmission 0.18 measure start 181 duration 88 transmission 0.20 measure start 361 duration 89 transmission 0.20 measure start 451 duration 89 transmission 0.19 end

Note that at day 10 the definition seeds an additional 300 infected persons onto the grid. This is done to make first and second wave similar in the way they utilize the whole grid (otherwise the first wave is very different in its geometry from the second one).

Running the file should give something like this:

 An example of a seasonal epidemic

Even given that the probability is never reduced any more folliwing 540, the disease never resurges after the second wave because the grid is sufficiently immune after the two waves.

## Waves by more contagious pathogen strains

As we have seen in the tutorial on Disease strains, a more contagious strain can either circumvent a containment measure that works for the original strain, or it can utilize the grid in the way the original strain could not.

Change the measures and seeding sections of the above definition as follows:

 seedings num_seedings 2 seeding time 10 number 300 transmission 0.20 seeding time 400 number 300 transmission 0.26 tag_index 2

This produces two waves of about equal height. Note that due to the increased transmission probability of the new strain, the second wave can be as steeply rising as the fist one which was not the case for the seasonal example.

 An example for a wave due to a more contagious pathogen.

A look at the spatial distribution of the two strains shows how the new strain manages to infect the 'pockets' left after the first wave expanded across the whole grid.

 Spatial distribution of the two waves, the more contagious strain (green) utilizing the gaps left by the original strain.

## Resurgent waves

A resurgent wave happens whenever the disease is slowed (or stopped) by measures but there is still too little immunity on the grid to prevent further propagation of the disease.

Dependent on details, the second wave may start immediately after measures are relaxed (when infections are still reasonably strong), after a long while (when infections are driven into a geometric situation where the advancing fronts find little non-immune people to continue) or only after an external perturbation (when the rate was driven to zero, but the situation is not yet stable).

The following file shows a situation in which a disease resurges two times after being driven down by containment measures, each time with less force.

 seedings num_seedings 1 seeding time 10 number 300 transmission 0.20 measures num_measures 2 measure start 50 duration 50 transmission 0.16 measure start 250 duration 50 transmission 0.16

Generally the wave after the second measure is developing rather slow, indicating that large parts of the grid already are immune.

 An example for a resurgent wave.

## Remarks

It follows from these experiments that there are many potential mechanisms which can lead to waves of a disease. In nature, the various factors also may mix. There may also be scenarios not considered here, for instance festive seasons which selectively lead to higher-than usual transmission probability because people engage in more risky activities than usual.

However, to understand and properly model the long-term waves of real epidemics, such as the plague in medieval times, or the yearly pattern of influenza, requires the crucial ingredient of how immunity fades over time for one or the other reason.

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