# Backward bifurcation and reinfection in mathematical models of tuberculosis

Wangari, I 2017, Backward bifurcation and reinfection in mathematical models of tuberculosis, Doctor of Philosophy (PhD), Science, RMIT University.

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Title Backward bifurcation and reinfection in mathematical models of tuberculosis Wangari, I 2017 Mathematical models are widely used for understanding the transmission mechanisms and control of infectious diseases. Numerous infectious diseases such as those caused by bacterial and viral infections do not confer life long immunity after recovering from the first episode. Consequently, they are characterized by partial or complete loss of immunity and subsequent reinfection. This thesis explores the epidemiological implications of loss of immunity using simple and complex mathematical models. First, a simple basic model mimicking transmission mechanisms of tuberculosis (TB) is proposed with the aim of correcting problems that are often repeated by mathematical modellers when determining underlying bifurcation structures. Specifically, the model makes transparent the problems that may arise if one aggregates all the bifurcation parameters when computing backward bifurcation thresholds and structures. The backward bifurcation phenomenon is an important concept for public health and disease management. This is because backward bifurcation signals that disease will not be eliminated even when the basic reproduction number R0 is decreased below unity; rather, for the disease to be eliminated, R0 has to be reduced below another critical threshold. I provide conditions to find the threshold correctly.Secondly, the simple basic TB model is extended to incorporate epidemiological and biological aspects pertinent to TB transmission such as recurrent TB, which is defined as a second episode of TB following successful recovery from a previous episode. I study the conditions for backward bifurcation in this extended model that features recurrent TB. Mathematical techniques based on the center manifold approach, are used to derive an exact backward bifurcation threshold. Furthermore, both analytical and numerical findings reveal that recurrent TB is capable of inducing a new and rare hysteresis effect where TB will persist when the basic reproduction number is below unity even though there is no backward bifurcation. Moreover, when the reinfection pathway among latently infected individuals is switched off, leaving only recurrent TB, the model analysis indicates that recurrent TB can independently induce a backward bifurcation. However, this will only occur if recurrent TB transmission exceeds a certain threshold. Although this threshold seems to be relatively high when realistic parameters are used, it falls within the recent range estimated in the relevant literature.The second TB model is extended by dividing the latent compartment into two: fast (early latent) and slow (late latent) latent compartments, to enhance realism. Individuals in both early and late compartments are subjected to treatment. The proposed TB model is used to investigate how heterogeneity in host susceptibility influences the effectiveness of treatment. It is found that making the assumption that individuals treated with preventive therapy and recovered individuals (previously treated for active TB) acquire equal levels of protection after initial infection, and are therefore reinfected at the same rate, may obscure dynamics that are imperative when designing intervention strategies. Comparison of reinfection rates between cohorts treated with preventive therapy and recovered individuals who were previously treated from active TB provides important epidemiological insights. That is, the reinfection parameter accounting for the relative rate of reinfection of the cohort treated with preventive therapy is the one that plays the key role in generating qualitative changes in TB dynamics. In contrast, the parameter accounting for the risk of reinfection among recovered individuals (previously treated for active TB) does not play a significant role. The study shows that preventive treatment during early latency is always beneficial regardless of the level of susceptibility to reinfection. And if patients have greater immunity following treatment for late latent infection, then treatment is again beneficial. However, if susceptibility increases following treatment for late latent infection, the effect of treatment depends on the epidemiological setting: (a) for (very) low burden settings, the effect on reactivation predominates and burden declines; (b) for high burden settings, the effect on reinfection predominates and burden increases. This is mostly observed between the two reinfection thresholds, RT2 and RT1, respectively associated with individuals being treated with preventive therapy and individuals with untreated late latent TB infection.Finally, a mathematical model that examines how heroin addiction spreads in society is formulated. The model has many commonalities with the TB model. The global stability properties of the proposed model are analysed using both the Lyapunov direct method and the geometric approach by Li and Muldowney. It is shown that even for a four dimensional model, the use of two well known nonlinear stability techniques becomes nontrivial. When all the parameters of the model are accounted for, it is difficult if not impossible, to design a Lyapunov function. Here I apply the geometric approach to establish a global condition that accounts for all model parameters. If the condition is satisfied, then heroin persistence within the community is globally stable. However, if the global condition is not satisfied heroin users can oscillate periodically in number. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users, are the key mechanisms fuelling heroin epidemic proliferation. However, in the long term, relapse of heroin users undergoing treatment back to a heroin using career, has the most significant impact. Doctor of Philosophy (PhD) RMIT University Science Dynamical Systems in Applications Biological Mathematics Numerical Solution of Differential and Integral Equations Backward bifurcation Recurrent TB Hysteresis Preventive therapy Heterogeneity
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