Covid Transmission: Simple Mathematics Explain The Danger Of Crowded Situations
Prof. Kirthi Tennakone
Reducing density of crowds and movement in such situations and isolation of infected or those suspicious of exposure, effectively limit the progression of coronavirus infection. Imposition of such measures and education of the general public is an absolutely a must to contain the infection.
Simple mathematics fathomable to most, explains how and when the infection expands into epidemic proportions. Attempting to grasp this idea would be an opportunity for students as well as laymen confined to their homes to refresh math they had learnt in the school and clear the myth that mathematics is difficult.
The tutorial below provides the mathematics prerequisite needed to understand the problem.
Mathematics Tutorial
In mathematics numbers are denoted by letters of the alphabet. For example, the letter N could represent the population density (number persons per square kilometer) of Colombo or the price of one-kilogram rice in rupees. For simplicity, the product of two numbers, say N and M (N x M) is denoted by NM or equivalently MN and the ratio of N is to M by N/M. The reader may have heard the statement that some quantity N is proportional to another quantity M. This means, the ratio N/M is always the same irrespective of the values N and M that are allowed and when expressed as an equation, proportionality is expressed as N/M = k (a constant) or equivalently N = kM.
Figure 1. Exponential growth of a population. Plot of population density (N) vs
Many quantities we are familiar, change with time and the rate of change depends on the time at which you measure it. How do we measure the rate of change of some quantity, say the population N of a city? If the population at certain time t is N and after lapse of very small interval of time denoted symbolically as dt, it changes to N + dN, where dN denotes the increase in population. The quantity dN/dt measures the rate of increase of the population at time t. Frequently the growth of a population at time t is proportional to the population at that instant of time itself, represented by the equation, dN/dt = kN, where k is a positive constant. In such situations referred to as exponential growth, N varies with time as shown in the figure below.
Equipped with above background knowledge you will be able to follow the mathematical argument of the following section, which explain why the movement in a dense crowd poses a severe danger in the present situation of the risk of COVID.
Mathematical explanation
The primary mode of transmission of COVID is believed to be the release of virus carrying droplets by coughing or sneezing of infected individuals. These droplets disperse around a range of the order of one meter (L = 1/000 of a kilometer). Suppose in one day you have moved a total distance of D kilometers in a crowed of N persons per kilometer square meter and among them a number M per square meter are carriers of the COVID virus. Figure.1 below pictures the crowd, grey dots representing healthy people and black dots the carriers of the virus. It is clear from the figure, your critical range of potential exposure, extending a distance L to right or left has swept an area 2LD, shown by the contour bounded by broken lines and you are exposed to infected persons covering this area which is 2LDM.
Your chance of catching the disease in one day of roaming in the crowd is proportional to 2LDM and therefore expressible as 2LDMP, where P is a constant of proportionality. The number P takes into account the fact that, every one exposed to the virus would not get infected. As there are N persons per unit area in the crowed largely in excess of infected individuals, each moving similarly, the rate of growth of the infection in the population per unit area is N times 2LDMP, which is 2PLDNM.
