The Concept of R-naught (R0) for CORONAVIRUS

The Concept of R-naught (R₀) for CORONAVIRUS

Dr Himanshu Shekhar

Introduction: For epidemics, reproduction rate or contagious potential is expressed as number of persons, who will contract a contagious disease from 1 person. This is mentioned as R-naught or R₀ in literature. Although it seems to be an important factor in calculation of spread or infections for a disease, but mathematical significance is belittled due to many lacunae in its definition, constituents, assessment and implementation. Most of the time, it becomes a postmortem number or manipulated number. An explanation is attempted in this post for mathematical calculation.

Concept: R₀ is defined as contagious potential of a disease and number of persons, which are infected from a single person. If this number is less than 1, then spread of disease is controlled and if it exceeds 1, the diseased spreads manifolds and attains the shape of Pandemic. Many past disease-spreads are assigned some numbers. 1918 Pandemic was assigned a range of 1.4 to 2.8, 2009 H1N1 spread is assigned a range of 1.4 to 1.6. Many prevailing diseases are given approximate R₀ values.

Hepatitis C & Ebola	HIV & SARS	Mumps	Measles
2	4	10	18

It makes Measles most contagious disease and it spreads at a faster pace, as compared to others. For COVID-19, initially R₀ of 2.2 to 2.7 is assigned, which is increased to 5.7, making it more contagious now. There are certain assumptions associated with the concept of R₀.

Assumptions: The calculation or estimation of R₀ is very difficult and it has limited mathematical acceptance in calculation. But it is mentioned as one of the parameters for the epidemics and pandemics. For the calculation of R₀, 3 major assumptions are made:

1. No one is vaccinated

2. None had any history of contracting the disease earlier

3. No way to control the disease exists

It indirectly indicates a hypothetical situation, where disease is left to spread and create havoc in unabated fashion. The people are made vulnerable in terms of external and internal protection and no shielding against disease spread is also visible. Under such circumstances, if an infected person is left in a vulnerable crowd, the potential of disease to spread is a measure of R₀. Mathematically, these assumptions are improper and reasons for infecting entire crowd could not be established. This makes the assumptions flawed.

Another domain of mathematical complexity exists in assessment of duration during which a person can infect others. Another medical term, called incubation period is defined, which is equal to time between exposure to virus and appearance of visible symptoms. It is presumed that after appearance of symptoms, infected persons are quarantined or isolated and spread of disease is restricted. The duration of incubation period is again a vague term, which for COVID-19 is expressed as varying from 4.2 days to 14 days.

One more domain for concern is duration for which so called contact can result in an infection. There is certain duration for which a healthy human being is exposed to contaminant, to become infected. The duration is not dependent on exposure but the media of contamination also. HIV or Ebola cannot propagate faster because despite having higher R₀, their mode of transmission is body-fluid, exposure to which is difficult. Contrary to this, air-borne diseases with lower R₀ can propagate faster, because direct contact is not needed and air is a universal carrier.

One major concern is reproduction rate of a disease will be a function of demographical and geographical variation and inherent resistance of population for a particular disease. So, reproduction rate may be different in different domains. People of one region may be prone to a disease, while in other area, immunity will be better. It is a function of environment, food habit, culture, tradition, rituals and lifestyle.

One major hurdle in mathematical adaptability of R₀ is absence of any time parameter in its definition. The duration which a person takes to infect R₀ persons is not indicated or included in the definition. If a disease remains in a person for 14 days and makes him worthy of infecting others, and the person fails to come in contact with any other person, during these 14 days, then despite having very high value of R₀, the spread will be restricted.

As these restrictions and limitations makes R₀, a non-mathematical parameter, only, some mathematical outline is assigned to this term, so that some insight from reproduction angle can also be derived.

Mathematics of R₀: There are three major mathematical constituents of R₀. First is infection period, which is the time during which an infected person can transmit the disease to other person. For common flue, it is 8 days. Longer the infection period, deadly will be the disease and more spread is expected. COVID-19 can be assigned an infection period of 14 days. Second parameter is Contract rate, which is defined as number of person contracted in a given time. Normal contract rate in India is of the order of 25-30 persons per day. In addition to this, everyday new persons are not contracted but there are repetitions and only 1% of new contracts can be assumed for a longer period of calculation. During correctly implemented lockdown, this contract rate is reduced to 5-6 persons per day, so as to reduce the multiplication of infections. Third parameter is mode of transmission. No numerical value can be assigned to this parameter but transmission through air is the fastest and easiest, if any disease has such potential. For COVID-19 spread, 1 can be assigned to this parameter.

So,

Without lockdown, the infection propagation rate is 14x25x1/100 = 3.5 persons.

With lockdown, the infection propagation rate can be 14x5x1/100 = 0.7 persons.

As per calculation, this infection propagation rate may be taken as equivalent of R₀.

Another approach is calculation of R₀ from doubling period, assuming infection period (IP) of 14 days and mathematically R₀ = 2^(IP/DP). For different values of doubling period (DP), the value of R₀ can be calculated.

Conclusion: A mathematical analogue for reproduction rate or R₀ is derived and its mathematical significance is established. An analogous infection propagation rate is mathematical established. The value of reproduction rate is correlated with doubling period also and for a doubling period of 14 days, R₀ is 2.0.It is also worth mentioning that the value of reproduction rate is not constant, but it changes dynamically with time and geographical region.

(Please note that this is a purely mathematical exercise and non-achievement of R₀ = 0.7 persons during lockdown is not contested or established by this post. This post depicts ideal situation and not reality.)

Dr Himanshu Shekhar