A self-satisfactory understanding on how COVID-19 spreads
Here is my self-satisfactory and yet mathematical understanding on how COVID-19 spreads and disappears. This is not specific to COVID-19, but it is general.
Let me assume that the number of infected persons per area n increases proportionally to the number of contacts per area per time Z between infected persons and potentially infected persons while it decreases proportionally to n. So the basic equation is written as
When people moves randomly like gas molecules [1], Z is proportional to n and the number of potentially infected persons n’. It is probably not so arrogant to assume n’ as N-n-V, where V is the number of persons with antibody per area, that is, the number of vaccinated persons per area V₀ and that of recovered persons per area. Therefore, Eq. (1) is written as
where
Basic reproduction number R₀ [2] that is commonly used in this field is defined as
where T is the average infection period. According to the definition of T,
In summary, Eq. (2) is described as
where r=n/N and v=V/N. This equation becomes
Here non-dimensional time 𝜏 = t/T is introduced, Eq. (7) is expressed as
where
These equations are cool because all of the constants and variants are non-dimensional. That means a very general I can say they are general. At initial stage of pandemic (Should I say epidemic?) at 𝜏 ~ 0, that is, when r ≪ 1 and v ≈ v₀,
Since the right term of Eq. (10) is constant × r, it is easily solved as
This solution is valid when r≪1. It is clear that r increases when 1-v₀-1/R₀>0 and vice versa. When v₀=1–1/R₀ (critical number), r stays constant. This critical number is called a herd immunity threshold [2]. In order to avoid pandemic (epidemic?), the (pre-)vaccination ratio v₀ must be larger than 1–1/R₀.
Anyway, I solved Eq. (8) and Eq. (9) with a spread sheet. The following chart shows how viruses spread and disappear when R₀=4, v₀=0 and r(0)=10⁻⁶ are assumed.
As a result, I found that rmax~0.4 at 𝜏~5. I can not estimate rmax or its corresponding 𝜏 by hand unfortunately. I can not find any indication at initial stage at 𝜏<3. But when I use a semi-log chart instead, things look differently.
In this semi-log chart, it is clearly shown that the infection rate r increases exponentially at initial stage (as predicted) and decays also exponentially at final stage. If I assume a different small enough r(0), I can estimate the behavior with a lateral shift of the plot. The behavior at the final stage in this case is expressed with a simple exponential decay like exp(-𝜏) since r+v≈1 (r≈0 and v≈1). The decay is slower roughly at R₀<3 when r+v does not reach close to 1.
How to reduce the infection from the view point of gas dynamics
When you look at Eq. (7), you can easily understand that R₀/T is very important. As its definition in Eq. (4),
According to general gas dynamics, α is proportional to infection sensitivity α₀, cross section σ of the contact (collision) and average relative velocity u. α₀ can be reduced by masks and u can be reduced by decreasing temperature. It is noteworthy that u is a relative velocity, so the velocity itself can be large. For example, in supersonic free jet, the average molecule speed exceeds the sonic. But all the molecules move to the same direction with little variation of speed. Therefore, the collision frequency between molecules is very low. A similar situation can be obtained in human behavior. If you have to move, move at the same velocity with others nearby. Take one-way route instead of both-way route. In molecular gas dynamics, the variation of molecular velocity is defined with temperature. The lower the temperature, the variation decreases.
References:
[1] W. G. Vincenti and Charles H. Kruger, “Introduction to Physical Gas Dynamics”
