Modelling infectious diseases with herd immunity in a randomly mixed population (preprint)

ABSTRACT

Background The conventional susceptible-infectious-recovered (SIR) model tends to overestimate the transmission dynamics of infectious diseases and ends up with total infections and total immunized population exceeding the threshold required for control and eradication of infectious diseases. The study aims to overcome the limitation by allowing the transmission rate of infectious disease to decline along with the reducing risk of contact infection.  Methods Two new SIR models were developed to mimic the declining transmission rate of infectious diseases at different stages of transmission. Model A mimicked the declining transmission rate along with the reducing risk of transmission following infection, while Model B mimicked the declining transmission rate following recovery. Then, the conventional SIR model, Model A and Model B were used to simulate an infectious disease with a basic reproduction number (r0) of 3.0 and a herd immunity threshold (HIT) of 0.667 with and without vaccination. The infectious disease was expected to be controlled or eradicated when the total immunized population either through infection or vaccination reached the level predicted by the HIT. Outcomes of simulations were assessed at the time when the total immunized population reached the level predicted by the HIT, and at the end of simulations. Findings All three models performed likewise at the beginning of the transmission when sizes of infectious and recovered were relatively small as compared with the population size. The infectious disease modelled using the conventional SIR model appeared completely out of control even when the HIT was achieved in all scenarios with and without vaccination. The infectious disease modelled using Model A appeared to be controlled at the level predicted by the HIT in all scenarios with and without vaccination. Model B projected the infectious disease to be controlled at the level predicted by the HIT only at high vaccination rates. At lower vaccination rates or without vaccination, the level at which the infectious disease was controlled cannot be accurately predicted by the HIT.   Conclusion Transmission dynamics of infectious diseases with herd immunity can accurately be modelled by allowing the transmission rate of infectious disease to decline along with the combined risk of contact infection. Model B provides a more credible framework for modelling infectious diseases with herd immunity in a randomly mixed population.