Model description

We developed a deterministic mathematical model of the transmission of DS and MDR-TB strains between the following mutually exclusive compartments: susceptibles St, uninfected individuals who are susceptible to TB infection; those exposed to TB and latently infected Lt, representing those who are infected and have not yet developed active TB; infectives It, comprising individuals with active TB who are infectious; the recovered Rt, who were previously infected and successfully cleared the infection through treatment or natural recovery. We use the subscripts s and m to denote DS-TB and MDR-TB quantities respectively.

The total population size, Nt, is given by

N is assumed to be constant and individuals mix randomly.

To assure the population size remains constant, we replace all deaths as newborns in the susceptible compartment. This involves death through natural causes, which occurs in all states at the constant per-capita rate μ, and TB-related deaths, which happen at the same constant per-capita rate ϕ for individuals in the Is and Im compartments. Individuals may also return to the susceptible compartment following recovery at the constant per-capita rate ω. Individuals enter the susceptible compartment at a constant rate μ through birth, where they may be infected with a circulating MTB strain at a time-dependent rate λit=βIit³⁸. Here, β is the probability of a susceptible individual being infected with MTB strain ii=s,m by an untreated infectious individual per day³⁸. A proportion βIstSt and 1-cβImtSt of the MTB susceptible individuals move to the latently infected compartment Lst and Lmt respectively. Here, c represents the MDR-TB fitness cost. We assumed that MDR-TB is initially generated through the inadequate treatment of DS-TB and could subsequently be transmitted to other individuals.

A proportion of latently infected individual’s progress to active TB as a result of endogenous reactivation of the latent bacilli at rates α. Moreover, since latently infected individuals have acquired partial immunity which reduces the risk of subsequent infection, a proportion also move to the susceptible compartment St at the constant per-capita rates ηii=s,m. This rate can be accelerated by treatment of latent TB. Some infectious TB cases will undergo spontaneous recovery at a rate γ, while others will die from TB-related causes at a rate,φ. The remaining individuals with drug-sensitive and MDR active TB Iit will eventually be detected and treated at rates δ and τii=s,m respectively. A proportion δτs of the treated DS active TB recover to move into the recovered compartment Rt, and a proportion of amplification ρ develops multi-drug resistance due to incomplete treatment or lack of strict compliance in the use of first-line drugs (drugs used to treat the DS forms of TB) to move into compartment Imt.

To confirm MDR-TB we need to do extra drug-susceptibility testing κ, therefore a proportion δκτm of MDR active TB cases recover to move to the recovered compartment Rt. Furthermore, a proportion γsandγm of individuals in compartments Ist and Imt naturally recover into Rt. A per-capita rate ω from the recovered compartment Rt move into the completely susceptible compartment St due to the loss of immunity. The model flow diagram is presented in Fig. 2.

From the aforementioned, the system dynamics are governed by the following deterministic set of nonlinear ordinary differential equations: