The third wave of Covid-19 is surging in India and the cases are increasing rapidly, but with lesser severity. From mid-November, the Omicron variant had started in Hong Kong and China and then gradually spread to the United States and up to 40 countries it has been detected. There are more than 50 mutations carried by the Omicron variant and only a few have been studied earlier. But it is very important to currently know in the third wave as well how fast the variant is spreading by estimating the basic reproductive number and also computing the required herd immunity of the population.
In this article compartmental deterministic mathematical model has been considered as India has a huge population. Here the individuals of the population are assigned to different compartments, each representing a specific state of the pandemic. The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. In this model, we are assuming that the total population is exposed to the disease looking at the current scenario. The total population N is exposed to the Omicron strain. The individuals are likely to be infected by the infectious persons in case of contact except those who are quarantined, have no travel history, take precautions, and are not in contact with a COVID-19 positive person. Those who remain undetected or are late detected or ignore their symptoms are the ones contributing to the disease transmission and spread, and those detected are isolated to the hospital for immediate treatment. The ones who recover have chances of becoming immune to the virus as the body has already developed antibodies to fight the virus and also the person is educated and cautious about the transmission of the disease. The death rate due to Omicron is denoted by 𝝁. Hence, N is the rate at which individuals are born into the susceptible class without any immunity and is the rate at which they leave it through death. The rate at which the susceptible class changes is equal to the rate at which infections occur, which takes place when the disease is passed from an infective individual to a susceptible one. The difference between the susceptible and the infective contacts is
proportional to the product of S(t) and I(t). The SEIR diagram below shows how individuals move through each compartment in the model.
Hence the rate of change in the susceptible individuals is
dS/dt = 𝝁N- 𝝁S-(βSI/N)
where βSI is the rate of infection
The term βSI/N is negative because the number of susceptible individuals goes on decreasing. The rate at which the individuals leave the susceptible population is equal to the rate at which they enter the exposed population. The number of individuals in the exposed class increases since those in the susceptible class become exposed to the disease. Let εE be the rate at which an exposed individual becomes infectious. Then the rate of change of the exposed population is given by
dE/dt = (βSI/N)-( 𝝁+ε)E
Let 𝛾I be the rate at which an infected person may recover. Then the rate of change of the infected population will be given by
dI/dt = εE –(µ+𝛾)I
The rate of change of the recovered is given by
dR/dt = 𝛾I-µR
This non-linear system of differential equations has the initial conditions
S(0)=S0>0, E(0)=E0>0, I(0)=I0>0, R(0)=R0>0
Also, the rate of contact (β), the rate of infection (ε), the recovery rate (𝛾) and the birth and death rate (µ) are all non-negative i.e., β>0, ε>0, 𝛾>0 and µ>0. Thus we have
Since (dN/dt)=0, we thus have N=S+E+I+R as a constant
The death rate in India due to COVID-19 as observed up to 9th January 2022, shows 1.34 deaths per 100 population.
The susceptible individuals who are not fully vaccinated are 1054525612 and the total population who are exposed to the disease and said to be taking necessary precautions including the ones vaccinated to the disease are N=1402294697
Therefore, transmission rate, β= (1054525612 /1402294697) = 0.752
The incubation period for COVID-19 is approximately 2.9 or 5.2 days (Kucharski et al, 2020) of being exposed. For the Omicron variant too somewhat similar incubation period is observed.
Therefore, the rate of infection may be given by
ε1= 1/incubation period = 1/2.9= 0.3448 per day
ε2= 1/incubation period = 1/5.2= 0.1923 per day
The expected duration of infection is the inverse of the removal rate (Jones J. H., 2007). For COVID-19, the recovery period is about 2 weeks i.e. 14 days which actually includes the incubation period as well. So, we assume a recovery period of about a week of being tested and hospitalized. Hence the recovery rate is
𝛾=1/recovery time =1/7= 0.1428 per day
Thus our estimated reproductive number is
When the incubation period is 2.9, R01= 4.6342
When the incubation period is 5.2, R02= 4.5007
Using the obtained basic reproductive number, we can estimate a threshold for herd immunity. Herd immunity is when enough people in a population have immunity to infection to be able to effectively stop that disease from spreading.
H1 =1- (1/ R01) = 1-(1/5.013)= 0.7842
H2=1 – (1/ R02) = 1-(1/4.7401)= 0.7778
Since R0 values are 4.6342and 4.5007, we can say that on average a COVID-19 positive person infects approximately 5 more people, as per the current scenario. It has been estimated that approximately 78% of the population has to be immune with recommended faster vaccination, in this case, to be infected in order to be immune to the disease, before the chain of transmission is interrupted.
Kucharski, A. J., Russell, T. W., Diamond, C., Liu, Y., Edmunds, J., Funk, S., … & Flasche, S. (2020). Early dynamics of transmission and control of COVID-19: a mathematical modeling study. The Lancet infectious diseases, 20(5), 553-558.
Jones, J. H. (2007). Notes on R0. California: Department of Anthropological Sciences, 323, 1- 19.