Fun

Misc

Pandemic

...it was in 2021

Preamble

1. It is not for the first year that the COVID-19 / SARS-CoV-2 pandemic has become a common topic of interest to all. It is in the news, government meetings, gossip and rumors, conspiracy theories... Conflicts between “masks supporters” and “anti-maskers”, those who are supporting vaccinations and those who do not are transformed into demonstrations and not always peaceful actions. Now is the 21st century, science has proven its worth, the development of medicine wiped out many terrible diseases. Amazing increase in average life expectancy was reached in just last hundred years (compare the data for the 19th century, the beginning of the 20th century and modern ones). However, when it comes to pandemic and vaccination, people often forget what they were learned in school. They refer to rumors or isolated cases when it is necessary to talk about statistics. More often they say “believe” / “do not believe” than refer to objective verified scientific data.

2. Once, at my leisure, an idea arose: why not try to build a model of a pandemic-epidemic? Such a model can be useful, well, at least interesting. Any model is a simplification, the accuracy of the model is always a subject of discussion. However, some interesting regularities and correlations can be obtained using even a simple model.

And below is what I got.

Model

The “mechanical part” of the (Monte Carlo) model is a simplified model of (two-dimensional) Brownian motion. Balls move without friction inside the rectangle. Interaction occurs only when the balls collide with each other and / or with the border of the rectangle. The interaction is described as an elastic collision. The initial position, speed and direction of movement of each ball are random values (in certain limits).

"Mechanical" parameters of the model:

the diameter of each ball is 1 (constant),
the size of the rectangular area (where the balls move) - in the diameter of the ball,
minimum and maximum speed of the balls,
number of balls
simulated time (of movement of the balls).

In addition, each ball has its own state: either “healthy”, or “sick”, or “recovered / vaccinated”, or “dead”.

Corresponding model parameters:

the probability of a “healthy” ball to become “sick” (upon collision with a “sick” ball),
the probability of a “recovered / vaccinated” ball to become “sick” (upon collision with a “sick” ball),
the probability of a “sick” ball to “recover”, that is, to become “vaccinated” during the entire simulated time,
the probability of a “sick” ball to “die” during the entire simulated time.

The change in the state of the balls occurs randomly in accordance with the given probabilities (Monte Carlo method).

The above parameters were chosen to numerically represent the “generally accepted” characteristics.
So the number of balls and the size of the region characterize the density of the “population”. Together with the velocities of the balls, this density characterizes the number of collisions (contacts) per unit time. Ball velocities, area sizes and simulated times also determine how many times the ball / participant (potentially) can cross the simulation region (travel distance or mobility). The probabilities speak for themselves.

Of course, the parameter values should be within reasonable limits. Otherwise, it is difficult to expect reasonable results 😉.

Simulation results

Mobility / number of contacts

First, let's see how the speed of the balls affects the results of numerical experiments.
At the beginning of each experiment, there were 10,000 "healthy" and 1,000 "sick". The general parameters of the model (probabilities, area size, velocities) are shown in Fig.1.
The red rectangle marks the parameters that were changed (in the first series) from experiment to experiment.

In four (computational) experiments, the minimum and maximum speeds were changed. That is, cases from very high mobility (the path traveled by one participant in the simulated time exceeds the size of the region by more than 5 times) to limited one (the path traversed by one participant in the simulated time is less than the size of the region by at least 4 times) were modeled separately.

Fig. 1.

Simulation results clearly show that the rate of morbidity and the peak values of the number of cases strongly depend on the speed of movement, and hence on the number of contacts. An increase in mortality is also evident with an increase in the number of contacts during the simulated time.

It was also shown that restricting movement only for a part of the population does not lead to a significant improvement in the situation. The increase in the number of patients and the number of recoveries is determined - mainly - by the participants who have not been affected by the restriction of movement.

Thus, movement restrictions significantly reduce both the risk of illness
and the (potential) workload of hospitals.

2. Vaccination

Now let's look at the impact of vaccination on morbidity, number of healthy people and deaths. The ratio of the numbers (initial values) of “vaccinated” and “healthy / never sick” were changed in numerical experiments, leaving their total number constant (10000).

Simulation results show that a noticeable effect of vaccination is appears only when the number of vaccinated reaches 50% (at least, more is better). At the same time, a larger number of contacts / high mobility (speed of movement) reduces this effect (of vaccination).

Conclusion

So, the unexpected did not happen. As common sense suggests, limiting the number of contacts and getting vaccinated can reduce sickness rate, morbidity (especially recurrence) and mortality. It should be especially noted that the effectiveness of these measures is highly dependent on the number of people involved / covered. Restricting movement and / or vaccinating only a small portion of the population does not improve the situation.

It is clear that it is impossible to force everyone to stay at home. And the economy will not stand it either. However, the number of contacts should be limited as much as possible for as much as possible people, and majority of population must be vaccinated to win over pandemic with minimum losses.

Addendum 1

Mobility / number of contacts

In four (computational) experiments, the minimum and maximum speeds were changed. That is, cases from very high mobility (the path traveled by one participant in the simulated time exceeds the size of the region by more than 5 times) to limited one (the path traversed by one participant in the simulated time is less than the size of the region by at least 4 times) were modeled separately. The values obtained at the end of the simulations are shown in Table 1.

Even from these data, one can see a strong dependence of the number of healthy, sick, dead from the number of contacts. A more detailed analysis can be carried out by considering the data in progress, that is, depending on time.

Fig. 2. Quantity of never-sick , recovered and their sum ( healthy ) vs the simulated time. The numbers (1,2,3,4) indicate the IDs of experiments (in Table 1)

Fig. 3. Quantity of sick , and sick 2 or more times vs the simulated time. The numbers (1,2,3,4) indicate the Ids of experiments (in Table 1)

Fig. 4. Quantity of deaths vs the simulated time.
The numbers (1,2,3,4) indicate the IDs of experiments (in Table 1)

Simulation results clearly show that the rate of morbidity and the peak values of the number of cases strongly depend on the speed of movement, and hence on the number of contacts. An increase in mortality is also evident with an increase in the number of contacts during the simulated time.

The movement speed was (almost) the same for all participants in the first numerical experiments.
Let's see what happens if the speed of movement is significantly different for different participants in the same experiment. That is, restrictions on movement will affect only part of the population. Below are the results for two experiments: minimum speed = 1, maximum speed = 50 , and minimum speed = 1, maximum speed = 30 . For comparison, the results of the experiments from Table 1 are shown.

Fig. 5. Quantity of never-sick , healthy (never-sick + recovered) , sick , sick 2 or more times and died vs the simulated time. Thick curves for speeds 1-50 . Thin curves for experiments 1 and 2 (in Table 1)

Fig. 6. Quantity of never-sick , healthy (never-sick + recovered) , sick , sick 2 or more times and died vs the simulated time. Thick curves for speeds 1–30 . Thin curves for experiments 2 and 3 (in Table 1)

It is clearly seen that restricting movement only for a part of the population does not lead to a significant improvement in the situation. The results in Fig. 5,6 show that the increase in the number of patients and the number of recoveries is determined - mainly - by the participants who have not been affected by the restriction of movement.

Addendum 2

Vaccination

Now let's look at the impact of vaccination on morbidity, number of healthy people and deaths. The ratio of the numbers (initial values) of “vaccinated” and “healthy / never sick” were changed in numerical experiments, leaving their total number constant (10000). The simulation results are presented below in Tables 2-4. The first row for the results of a "pure" experiment (10000 healthy, 0 vaccinated, 1000 sick); second row: 9000 healthy, 1000 vaccinated, 1000 sick; third row: 8000 healthy, 2000 vaccinated, 1000 sick; fourth row: 5000 healthy, 5000 vaccinated, 1000 sick; fifth row: 2000 healthy, 800 0 vaccinated, 1000 sick .

The dependencies of the number of healthy, sick and dead on time for all experiments are shown in Fig. 7-12.

Fig. 7. Quantity of healthy (never-sick + recovered) , sick , sick 2 or more times vs the simulated time. The numbers on the curves correspond to the row Id in Table 2.

Fig. 8. Quantity of deaths vs the simulated time. The numbers on the curves correspond to the row Id in Table 2.

Fig. 9. Quantity of healthy (never-sick + recovered) , sick , sick 2 or more times vs the simulated time. The numbers on the curves correspond to the row Id in Table 3.

Fig. 10. Quantity of deaths vs the simulated time. The numbers on the curves correspond to the row Id in Table 3.

Fig. 11. Quantity of healthy (never-sick + recovered) , sick , sick 2 or more times vs the simulated time. The numbers on the curves correspond to the row Id in Table 4.

Fig. 12. Quantity of deaths vs the simulated time. The numbers on the curves correspond to the row Id in Table 4.

The data above show that a noticeable effect of vaccination appears only when the number of vaccinated reaches (at least) 50%. At the same time, a larger number of contacts / high mobility (speed of movement) reduces this effect.