With the recent COVID-19 pandemic, several people are shocked to see the rapid progression of the number of COVID-19 cases. Beyond the spread occurring cross-borders, the inter-country disease propagation is also something that is quite shocking. In order to understand the growth in cases, it is paramount to understand a couple of fundamental principles and how these apply to the mechanics of disease spread.

The first principle is epidemiological compartmental modelling. Essentially a compartmental model is one that partitions the population into a set of independent states in relationship to the virus. The most simplistic and intuitive state partitioning is into susceptible (S) individuals, infected (I) individuals, and recovered (R) individuals. In this way, the breakdown forms the basis for a simple mathematical model called the SIR model. This is a foundational model that dates to the twelfth century [1]. By adding an additional incubation period to the model, we can further partition the population into an additional exposed (E) state, and we obtain an SEIR model. Most modelling efforts assume that this is a latent state and so it merely impacts the timing but not the extent of the spread of the virus.

Without delving into too many mathematical details, the spread of the virus is dependent on one critical rate, which is the reproductive rate (something called reproductive number), often denoted as !! [2]. In general, this term refers to the number of individuals that a person with the disease will infect. Therefore, if an individual is infected and subsequently infects 2 people, the reproductive rate in this case would be !! = 2. On the other hand, if 2 people have the virus and only 1 of these people passes it to 1 individual, then the reproductive number is !! = 0.5. In epidemiological modelling, this reproductive number is directly proportional to another constant ', which is usually called the transmission rate [3].

The basis of the propagation of the disease-spread is guided by the law of mass action, which states that the number of new infections depends on the product of the number of infected individuals, the number of susceptible individuals, and a transmission parameter [4]. In a more mathematical sense, this means that per change in time, we obtain the following per-change in the infected population: '(). By contrast, this implies a reduction in the susceptible population (who are now infected) of −'(). If we further introduce a recovery

rate of + then we see that over time, infected individuals move from infected to recovered at a rate of +). The system of equations governing these dynamics are the simplest form of SIR models:

(̇ = −'())̇ = '() − +)!̇ = +),

where /̇ represents the time-derivative, or the rate of change of the state /.

The second principle is exponential growth. Exponential growth, put simply, occurs when the rate of change of something is proportional to itself. Processes like compound interest are modelled through exponential growth. In the above, SIR model we focus on the infected state and recognize that the term '() represents a quantity that is proportional to ). Assuming that the susceptible population is large enough to nearly represent a constant volume (at the beginning of the disease propagation) we see that essentially this means that for some constant 0 we get:

)̇ = 0). Using )(0) = )! to represent the initial infected population, this is a differential equation whose solution is:

)(3) = )!e"# , which represents exponential growth. The dangerous thing about exponential growth is its ability to rapidly multiply even when it seems at first like subtle progression. Comparing some basic functions – linear, power, and exponential – we see below an example in the short term and slightly longer term:

In the short term, it appears that exponential growth begins a bit of a slow growth; however, we can see how it begins to rapidly increase later. This is exactly what we see when we look at the case numbers – slow starts, followed by rapid increases in cases.

The main point is that, without mitigation strategies, exponential growth of cases is to be expected. It is not unusual to see the surges in caseloads that we are experiencing both within several countries, and inevitably, globally. However, we’re already beginning to see positive impacts of social distancing and have observed in other countries how these strategies can have a substantial impact on flattening the increase in cases or globally reducing the caseloads. A follow-up post will show how different strategies act to change the dynamics in the mathematical models explored with a note of how social distancing and other mitigation strategies can fundamentally reduce, flatten, or completely halt the progression of the virus.

References

[1] R.M. Anderson. Discussion: the Kermack-McKendrick epidemic thresh-old theorem.Bulletin of mathematical biology, 53(1):1–32, 1991

[2] Heffernan, J.M., Smith, R.J. and Wahl, L.M., 2005. Perspectives on the basic reproductive ratio. Journal of the Royal Society Interface, 2(4), pp.281-293.

[3] Li, M.Y., Graef, J.R., Wang, L. and Karsai, J., 1999. Global dynamics of a SEIR model with varying total population size. Mathematical biosciences, 160(2), pp.191-213.

[4] Grassly, N.C. and Fraser, C., 2006. Seasonal infectious disease epidemiology. Proceedings of the Royal Society B: Biological Sciences, 273(1600), pp.2541-2550.

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