Mathematical modelling of the tuberculosis epidemiology / Ayinla Ally Yeketi - Ayinla , Ally Yeketi (2020)
This project analyses the tuberculosis (TB) epidemic mathematically using compartmental modelling approach. Three models are presented to discuss drug susceptible and multi-drug resistant TB. The first model presented has 4 compartments; susceptible, exposed, infectious and recovered. The relevance of the exposed class in managing TB is analysed and found to be useful in delaying the eventual onset of the infection. Compared to previous researches, our results significantly show that when efforts are made such that no infected individual bypasses the exposed class and progresses directly to the infectious, the TB epidemic is successfully combatted. Also, the model is used to show that reinfection has no significant relevance to TB incidence. The model is further extended to accommodate vaccination compartment. The vaccination compartment is included to understand the usefulness of a prophylactic vaccine in curbing the growth of TB epidemic. The intrinsic features to be considered in the formulation of the vaccine as well as the effective proportion of people to be vaccinated to achieve herd immunity are presented. A vaccine that would combat the infectivity rate of TB by half displays its potency in drastically reducing the TB incident rate. Placing the patient on good diets also gives a better result as discussed in the optimal control section. Furthermore, a model to analyse the relevance of quarantine in managing the incident rate of multi-drug resistant TB (MDR-TB) is formulated. The quarantine compartment is created to harbour individuals that develop MDR-TB. This shows its efficacy to help; monitor the recovery rate of individuals with MDR-TB, keep the MDRTB patients under watch regarding their medications, and as well prevent the MDR-TB patients from mingling with susceptible individuals. This model is shown to undergo backward bifurcation which gives a vital information on how to deal with the epidemic. In general, the equilibria points of all the 3 models are shown to be locally asymptotically stable whenever the basic reproduction number is less than unity (