Tuberculosis reinfection rate as a proportion of total infection rate correlates with the logarithm of the incidence rate: a mathematical model

Abstract

In a significant number of instances, an episode of tuberculosis can be attributed to a reinfection event. Because reinfection is more likely in high incidence regions than in regions of low incidence, more tuberculosis (TB) cases due to reinfection could be expected in high-incidence regions than in low-incidence regions. Empirical data from regions with various incidence rates appear to confirm the conjecture that, in fact, the incidence rate due to reinfection only, as a proportion of all cases, correlates with the logarithm of the incidence rate, rather than with the incidence rate itself. A theoretical model that supports this conjecture is presented. A Markov model was used to obtain a relationship between incidence and reinfection rates. It was assumed in this model that the rate of reinfection is a multiple, ρ (the reinfection factor), of the rate of first-time infection, λ . The results obtained show a relationship between the proportion of cases due to reinfection and the rate of incidence that is approximately logarithmic for a range of values of the incidence rate typical of those observed in communities across the globe. A value of ρ is determined such that the relationship between the proportion of cases due to reinfection and the logarithm of the incidence rate closely correlates with empirical data. From a purely theoretical investigation, it is shown that a simple relationship can be expected between the logarithm of the incidence rates and the proportions of cases due to reinfection after a prior episode of TB. This relationship is sustained by a rate of reinfection that is higher than the rate of first-time infection and this latter consideration underscores the great importance of monitoring recovered TB cases for repeat disease episodes, especially in regions where TB incidence is high. Awareness of this may assist in attempts to control the epidemic.