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Phase transition in Kermack-McKendrick Model of Epidemic: Effects of Additional Nonlinearity and Introduction of Medicated Immunity

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, the 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 or quadratic powers of the individual variables and introduce two new parameters to incorporate further nonlinearity in the number of infected people in the model. As a result of this, we show how this additional nonlinearity, in the newly introduced parameters, can bring a significant shift in the peak time of infection, i.e., the time at which the 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 it as a problem of phase transition. Then, we investigate all the necessary parameters of this phase transition, like the order parameter and critical exponent. We observe that $O_p \sim (q_c-q)^{\beta}$. {\it As far as we know the phase transition and its quantification in terms of the scaling behaviour is not yet know in the context of pandemic}. 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.

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