By 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.
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.
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.
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.
The infected persons are also removed at a rate proportional to their density in the population owing to immunity, isolation or death, giving the removal rate as kM, where k is a constant. Therefore, the net rate (infected rate – removal rate) dM/dt of the increase of infected persons in the population per unit area can be written as, dM/dt = 2PLDNM – kM = (2PLDN – k )M
The above equation tells you, if 2PLDN is greater than k, the disease grows to epidemic proportions. Evidently, the more you move in a dense population, you provoke conditions favorable for induction of an epidemic. The epidemic can be contained by limiting crowding of people (reducing N) and movements in such environments (reducing D) and effective isolation of the infected (increasing k).
The average distance D a person walks in his social environment is around 4.75 km day and initially, the rate of removal of infected individuals is negligible compared to rate of infection (first term of equation written in the previous section). The important unknown P in this simple theory, deduced from the curve similar to Figure.1 for the initial epidemic in Wuhan China is of the order of 1/100.
How many people does one single infected individual pass the infection?
If an infected individual move around in a population of density N per km square to attend routine matters as usual and if D is the total distance moved during a day. His critical range 2L of possible passing of the infection will trace an area 2DL. As the number of people in this area is 2DLN, the number likely to catch the disease would be 2DLNP per one day. An infected individual continues to be contagious for about 14 days. Therefore the number of people he is likely to pass the infection is 28DLNP. When we set D = 4.75 km, L = (2 x 1/1000) km, N =1500, P =1/100, the above number turns out be nearly 3. An infected person associated with a population of density 1500 per square kilometer will transmit the disease to about 2 persons on average. When population density doubles this number is also doubled.
The above characteristic number “ How many people does one single infected individual pass the infection”, measuring the intensity of an outbreak is referred to as reproduction number or R0 by epidemiologists. George Macdonald – British physician cum mathematician noted the importance of keeping R0 below 1 to control an epidemic and suggested a successful strategy to eradicate malaria in Sri Lanka
The model described above is not applicable to a circumstance where number of healthy persons and few infected confines to closed environment. Here, unless adequate precautions are taken, the numbers infected will be much larger.
Mathematics indicates that the current recommendations for containing the pandemic, needs to be strictly adhered.
*Prof. K. Tennakone, National Institute of Fundamental Studies, Hantana Road, Kandy.
The author wish to thank Prof. Saman Seneweera, Director National Institute of Fundamental Studies (NIFS) for encouragement and Dr. Sanath Wijesinghe, Massachusetts, United States for suggestions and carefully reading the articles. Discussions with Prof. Lakshman Dissanayake (NIFS) and Prof.Ajith De Silva, University of West Georgia, United States were invaluable in drafting this note. He is indebted to Kasun Jayanath Wimalasena for checking the calculation and attending to the drawings.