Introduction: The risk of contact infection among susceptible individuals in a randomly mixed population can be reduced by the presence of immune individuals and this concept is referred as herd immunity. Although herd immunity is observed in vaccinated population for some infectious diseases, it has never been truly attained in compartmental models such as the susceptible-infectious-recovered (SIR) model. This paper introduces a new SIR framework to overcome the limitation of the conventional SIR model in attaining herd immunity.
Methods: Two SIR models were newly developed based on the reduced risk of contact infection. The first model A assumes that the risk of contact infection reduces as soon as susceptible individuals are infected and move from class S(t) to I(t), therefore incorporating prevalence of both infectious and susceptible individuals into its force of infection. The second model B assumes the risk of contact infection would reduce after infected individuals have recovered from infection and move from class I(t) to R(t), therefore incorporating the prevalence of infectious and the inverse of prevalence of recovered individuals into its force of infection. Then, numerical simulations were applied to obtain approximate solutions for all three conventional SIR model, new SIR model A and model B for comparison under exact and arbitrary conditions with β = 0.3 and σ = 0.1 to mimic the infection dynamics with basic reproduction ratio (r0) of 3.0 and herd immunity threshold (HIT) of 0.667 (66.7%).
Results and discussion: All three models performed likewise at the initial stage of epidemic. The conventional SIR model simulated the epidemic diminishing when 94.0% of the population had been infected and recovered, way above its HIT. Model A simulated the epidemic waning when 66.7% of the population had been infected and recovered, in line with its HIT, however, the model conceptualized the herd immunity incorrectly. Model B simulated the epidemic waning at 75.6%, slightly above its HIT and was in line with the fundamental of herd immunity. The difference between model A and model B can be attributed to the proportion of infectious individuals, and this would increase in infectious disease with high transmissibility. The threshold theorem derived based on r0 may not be sufficient for optimal control and eradication of infectious disease with high transmissibility like the COVID-19.
Conclusion: The newly developed SIR model that includes the inverse of proportion of recovered individuals into its force of infection is more accurate and credible for modelling infection with high transmissibility or vaccine-induced herd immunity in a randomly mixed population, especially in COVID-19.