Warning: This Article Contains Immutable Mathematical Facts and Science - Infectious Disease Transmission and the R0 Explained

Calculating The Spread of Infectious Diseases

by MTY - Research Scientist -

R0 number is called the basic reproduction number for a disease… the most important number describing how a disease spreads …

Here’s how it works:  An infectious individual enters a population (a room with people in it, for example).  The R0 number tells us how many people in that population will become infected by that one person.

The flu we commonly battle each winter in various forms has an R0 number around 1.5.  The R0 for Covid-19 has ranged from 2.5 (probably much too low) to well over 5.  Here’s an idea of how much R0 affects a population:

If a disease has an R0 of 2, it means an infected person entering a population where no one else has been exposed to the disease or is ill / has recovered from it will infect two people in that group.  In turn, EACH of those two people will infect two more (meaning 4 more), etc.

In five generations of contacts, that 1st person will have infected 2 x 2 x 2 x 2 x 2 = 32 individuals.

If R0 is 5, the 1st person will (in five generations) infect 5 x 5 x 5 x 5 x 5 = 3125 individuals.

This is how the steep curve that overwhelms the medical system comes about.


R0 is calculated in different ways, depending on the particular epidemiological model used.  

One of the simpler models came from a landmark 1927 study and became what is known as the SIR model… Susceptible / Infected / Recovered.  In the beginning, EVERYONE in a given population is considered susceptible (barring natural immunity).  For modeling purposes, those with immunity are disregarded.  Once someone with the disease enters the population, a certain number of susceptibles are infected (according to the R0 number).  They leave the susceptible class (formally called a compartment) and enter the infected class.  While they are infectious (depending on the disease, being infectious may not be the same as being infected… in other words, we might be sick but the period where we can infect others may be over) these individuals will infect others, thus removing those people from the susceptible class and placing them in the infected class. 

Eventually, depending on the disease, some of those infected will recover and leave the infected class … moving into the recovered class.

There are certain numbers associated with this model: one is called Beta, the probability that someone coming in contact with an infectious individual will become infected. 
Note: This is DIFFERENT from the R0 number.  (The explanation for how Beta is calculated is a little involved.)

Another important number is Gamma, the “reciprocal” of the average amount of time necessary to recover from the disease.   The average recovery time for Covid-19 is about 14 days.

In the SIR model, R0 is calculated by dividing Beta by Gamma. 


There are many other models: 

a)     SEIR: Susceptible – Exposed – Infected – Recovered.  This model extends the SIR model with the added exposed class of asymptomatic individuals.  During a period of latency, individuals in this compartment cannot infect susceptibles.  At the end of this period, exposed individuals enter the infected class before recovery with assumed permanent immunity.



b)    SEIRS: Susceptible – Exposed – Infected – Recovered – Susceptible.  Similar to theSEIR model, the additional S-compartment refers to an assumed temporary immunity, after which individuals re-enter the S-compartment.



c)     All of the above models can also be designed to reflect birth and death rates; i.e., with dynamic populations that change over time. 


In the original models above, a “closed” (unchanging) population was assumed.  At any given moment, the sum of the individuals in every compartment of a given model should equal the total population number.  At the beginning, ALL of the individuals are in the susceptible compartment before some become infected and move into the infected class.  Eventually, some of those recovered and leave the infected class.

In a model with a dynamic population, the total population varies over time so the sum of the individuals in all of the classes at a given time will vary from day to day.  Plus, in a dynamic population, deaths from both the disease and from other causes can also be factored into the model.  In other words, it can become more complicated.

The compartments themselves can also be broken down into sub-classes such as gender, ethnicity, or age groups.

An important note about the number Beta I mentioned above.  The probability of becoming infected is the contact rate for the infection… What’s the probability of becoming infected based on number of interactions with others?  This is why we socially distance because it lessens the chance of infection.  If I walk into Grand Central Station during rush hour on a normal day, I potentially come in contact with at least hundreds of people, depending on how long I stay.  Staying home for an extended period means I come in contact with no one on the outside, thus sending that contact rate to approach zero.  As Beta decreases while Gamma stays the same (the average recovery time is relatively stable), the R0 number drops dramatically.

When R0 reaches a value of 1, we say it is at the trans-critical bifurcation… the point where the disease can “go either way”… where it can barely exist as an epidemic or simply exist in the endemic state (It will not become extinct but will infect very few over time).  Covid will hopefully reach the endemic state over the next year or so.  A vaccine will certainly help this along, though that may be well off in the future.

We also notice R0 is time-dependent… mitigation factors can lower it greatly.  

Flattening the curve was done especially to stem the tidal flood of patients in hospitals.  In the beginning, the R0 number was very high and infected cases inundated the system.  Unchecked, this would have devastated the population… in NJ models, there were projections of about 3 million or so infected with mitigation factors that were put in place.  It really did make a difference. 

Cooking the numbers and opening up as though this virus has left is a horrible plan.  If we screw this up, we could be right back to where we started. That’ll put everybody out of business.