Remodelling Kermack-McKendrick Model of Epidemic: Effects of Additional Non-linearity and Introduction of Medicated Herd Immunity

Datta, Agniva; Acharyya, Muktish

Mathematical modelling of the spread of epidemics has been an interesting challenge in the field of epidemiology. The SIR Model proposed by Kermack and McKendrick in 1927 is a prototypical model of epidemiology. However, it has its limitations. In this paper, we show two independent ways of generalizing this model, first one if the vaccine isn't discovered or ready to use and the next one, if the vaccine is discovered and ready to use. In the first part, we have pointed out a major over-simplification, i.e., assumption of variation of the time derivatives of the variables with the linear powers of the individual variables and introduced two new parameters to incorporate further non-linearity to the number of infected people in the model. As a result of this, we showed how this additional nonlinearity, in the newly introduced parameters, can bring a significant shift in the peak time of infection,i.e., time at which infected population reaches maximum. We show that in special cases, even we can get a transition from epidemic to a non-epidemic stage of a particular infectious disease. We further study one such special case and treat as a problem of phase transition. Then, we investigate all the necessary parameters of this phase transition, like order parameter and critical exponent. We observed $O_p \sim (q_c-q)^{\beta}$. In the second part, we incorporate in the model, a consideration of artificial herd immunity and show how we can decrease the peak time of infection with a subsequent decrease in the maximum number of infected people. Finally, we estimate a critical value of the rate of vaccination by a statistical method such that we proposed a way of possible eradication of the epidemic in a short time by effectively providing the vaccine to a population.