Can the stochastic motion of kinetic theory of gases be a good mathematical model of diffusion of Infectious diseases?

The stochastic motion and the theory of gases as a tool of mathematical modelling the diffusion and spreading of Infectious diseases.

The spreading of diseases in the population is a stochastic process which depends on certain parameters. We attempt to model it through stochastic interactions and a modified theory of kinetic gases.

We adopt laws of kinetic theory of gases where the microscopic atoms and their collisions represent the humans and their interactions and by following basic physical principles we model how the disease spreads microscopically. We modify the physical theory to introduce the necessary new parameters and then we simulate the spreading of the disease.

Before we move on: Does the simulation capture the basic characteristics of real data?

It does and surprisingly well!

*** The actual data of infected individuals at a given time (https://www.worldometers.info/coronavirus/country/china/):

The simulated stochastic interaction data of infections at a given time. The purple curve is the simulated on corresponding to the aqua-colored curve of the above data (the numbers are not important on the comparison and depend on the parameters of the system):

Notice the asymmetry on the curve and the long decrease (tail) of the infectious cases both in real and simulated data.

*** The actual data of total deaths at a given time (https://www.worldometers.info/coronavirus/country/china/):

The simulated stochastic interaction data of total deaths at a given time:

Notice the similarity and the fact that the deaths compared to the number of active infections follow similar tendency with the actual data.

The relative quantitative comparisons do not matter here, they depend on the death rate, the time that the infection lasts and the activity of city on the stochastic models.

Outreach explanation of the Stochastic model

In its simplest form the physical model contains particles of the same mass, which they rapidly collide among themselves and the walls of the container with perfectly elastic collisions. The average kinetic energy represents how active is the city and how many times the people interact to each other. In the physical model this is translated to the temperature.

To be more precise the model parameters and the mapping between Epidemic Diffusion<---> Physical Microscopic Theory are:

a) Population density. <---> The density of the particles.

b) How active are the individuals, i.e how much social distancing is exercised; and how transmittable is the disease. Complete ideal lock down corresponds to a velocity that approaches to zero.<---> The velocity/the average kinetic energy of the particles which corresponds to the temperature of the physical system.

c) Time of individuals being sick and can transmit the disease. <---> No kinetic theory of gases counterpart, the atoms have to be labelled with time dependent properties.

d) Critical value of the density of the infectious individuals where the health system collapses i.e. does not have the ability to offer treatment to everyone. <---> No kinetic theory of gases counterpart the atoms have to be labeled with an additional property.

e) Probabilities of the infectious people to be cured and become immune or to pass away. When the health care system collapses the later probability increases. <---> No kinetic theory of gases counterpart, the labeled atoms have an additional time dependent property.

The Basic Reproduction Number R_0: In Mathematical epidemics terminology corresponds to the number or cases generated by a single infected individual when there is no immunity yet in the system, i.e. the whole population is susceptible. It is one of the most important parameters to be determined in the modelling; an epidemic disease has R_0>1 and the highest the number is the easier is the spread of the disease. In our stochastic formalism the basic reproduction number can be defined in several ways, the simplest definition is to consider a model where a collision leads to infection, and in this case the most naive correspondence would be that the R_0 is the collision frequency multiplied by the time that the disease last for each individual. The collision frequency depends proportionally on the density of atoms and on how active the system is (the mean velocity).

The Individuals in the model:

Blue: Healthy people, not infected; in Mathematical Epidemics terminology (MEt) called Susceptible.

Purple: Infectious.

Green: Immune.

Red: Dead.

Immune+ Dead: Removed in MEt.

Three snapshots of the population status during the simulated evolution of the disease are shown below: Beginning of the spreading; Intermediate Stage; Final Stage.

The spread of the infectious disease occurs in a homogeneous population, homogeneously mixed and closed. We start with a city of fixed number of individuals for the web simulation, let us say 400. A minimal number of them are infectious and we let the system develop. Interaction between a healthy individual(blue) and an infectious(purple) leads to infection of the healthy person. The rate of collisions depends on the velocity of the particles. The infected individual remains infected for a certain time and after that the he/she gets immune (green) with a large probability P1 or may pass away (red) with a smaller probability 1-P1.

Once there is a large number of infectious people that reach the critical density where the health system collapses, the P1 probability of recovery slightly decreases.

For the model is possible to obtain analytic formulas, but for the web version let us focus on the numerical simulations.

<Our coding for the simulations of the webpage has been done in python.>

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The Realistic Scenario for the Disease Spreading: Healthy - Infection - Recovery Immunity or Death. E.g. The Coronavirus 19.

The infected individuals can be cured and become immune or die with a much lower probability. For the web presentation we use simulations of few hundred individuals, for more quantitative results to be used for scientific purposes we will be exercising on bigger samples.

A population of average activity with medium measures against the disease spreading and mild social distancing generates the following stochastic simulation:

Where the final stage of the simulation is depicted in the following graphs:

Notice the very large percentage of individuals in the city that has become infected. This is depicted in the first blue curve, which shows the rate of individuals who have not been infected reduces rapidly. Only a small percentage was lucky enough to avoid infection as is seen at the tail of the curve.

In the second purple curve notice the shape of the peak which is quite wide and how after that this reduces following a less steeper decrease than the increase. This is important, the curve is not symmetric, the rate of decrease of the number of infected people is not equal to the rate of increase. This means that the disappearance of the disease requires more time to be invested and this should be taken into account when relaxing the measures.

The red curve depicts the number of deaths. We have set artificially the health care capacity line to this system to be 35. When the health care capacity is reached by the number of infected people the death rate increases from 3% in this system to 8%. The increase of rate of deaths happens with a relative delay(as in the real data) compared with the infectious curve and this is mainly due to the time required to conclude if an individual will become immune or pass away. There is a relative increase of the deaths in the system due to the fact that the number of infectious people pass the health care capacity.

Why herd Immunity itself is not the solution against prevention measures?

Notice from the above simulation, that while the city continues its usual operations only a small number of people is not eventually affected by the disease. The number of individuals that do not get infected is approximately only 5%. The herd immunity then protects only about 4-5% of the city while in this simulation the death rate turns out to be 7.5%. The herd immunity is not expected to work at all with acceptable results in a system that operates as usual and other prevention measures against the spreading are necessary. This is the lock down and the social distancing.

Why prevention measures social distancing or lockdown are necessary? A demonstration of social distancing of lockdown by reducing the density of active individuals:

Let us keep constrained half of the system's population which is a way to demonstrate social distancing. The simulation demonstrates an extreme flattening of the infectious curve, below the system's health capacity and result to a considerable decrease of the number of deaths! Social distancing have tremendous effects on the reducing the speed of the spreading too. Notice also the large number of individuals (blue) that have not been infected at all when the disease has gone, by initially enforcing prevention measures the herd immunity is expected to work much better after the relaxation of measures.

A demonstration of social distancing of lockdown by reducing the overall activity of active individuals:

Lets simulate an extreme social distancing where the individuals have reduced activity and therefore lower number of interactions. The curve is flattened, the fatality rate is much lower and the city returns to its usual operations after the disease disappears. Notice again the much higher number of individuals that have not been infected in the simulation.

What the Kinetic Theory of Epidemics can predict for the future?

Early relaxation of the measures on the maximum of the curve may lead to peaks that last longer and to even a second higher peak. This simulation assuming no cure and vaccination exist and the city returns to its usual activities when relaxing the measures. We like to predict how the infectious curve will behave by relaxation of prevention measures. In the following simulations the health care capacity is not significant and can be ignored more over the webpage simulations are oversimplified and contain relatively low number of statistics.

In the first scenario we double the activity of the city after the measures have been relaxed, just immediately after the first pick of the infectious curve. It leads to extension of the high peak period and in this scenario even to a higher second peak. This is a likely scenario for countries with first outbreaks that have low peaks and the disease has not been spread to a very large percentage of the population due to early prevention and lock down measures. Wrong evaluation of when the measures should lifted may lead to worst spreading and a second larger peak (the simulation does not take into account other factors like weather conditions etc. that may play role in the disease spreading).

In this following simulation we increase the activity of the city by ~60% by the relaxation of the lockdown measures. This is a more realistic scenario. We make it happen at the time just after the peak of the curve where the decline has already just started. A curve stays much longer around its peak and even produces a second peak. This scenario leads also to increased number of deaths. According to this simulation the relaxation of measures needs to be delayed at a much later stage in order to avoid extensive peak.

In this scenario we increase by 60% the activity of the system when we approach to the declined tail of the curve, when the decrease has been already reached at least to around 30% relative to the peak and around 130 simulation time. You can notice a minor peak at this time in the graphs due to the relaxation of measures, which does not affect the fate of the curve significantly. Therefore, according to the simulation this is a safe timing to relax the measures which does not lead to additional drastic peaks.

Two different Cities and their Exchange of Individuals

We consider two different cities with a minor exchange of citizens under strict control. The stricter the control of the residents transporting between the two cities the lower the possibility of the infection is. The possibility of the infection from one city to another is never zero as long as there is an exchange of individuals. Very strict exchange, corresponds to a very narrow passage that the particles have to cross. Notice that once the disease have been transmitted from city A to city B the disease spread in city B is almost as bad as the city A. Notice that in these simulations the quantitative results and predictions become more challenging due to the difficulty and complications to quantify the actual restrictions and controls the citizens exchange with the length of the passage in the simulation model.

In the following simulation the passage is almost the size of a single particle, there is an extremely low possibility due to stochastic phenomena that the infection can be transmitted from one city to another. However notice that even in the particular simulation an infectious individual manage to pass to the city B around simulation time 20, although he has no active interaction during the time that can transmit the disease and the city B remains clear.

The qualitative outcome of those simulations is not surprising. As long as there is an exchange of people between two cities where one has infected individuals, the possibility that the disease will be transmitted to the second city is never zero. However, according to stochastic modeling, with strict control measures it can become extremely low.

This article has been uploaded on my website on April 10 and by April 14 all the simulations have been added. You may use the information of this article but please mention the source appropriately and if you are using it for scientific purposes please cite the article and feel free to contact me.