Influence of the smear-microscopy in global dynamics of tuberculosis transmission

Tuberculosis, a lethal infectious disease attribute among the top 10 causes of death globally and leading cause of death from a single infectious pathogen (rank before HIV). The sputum smear-microscopy and chest X-ray are the key TB diagnosis methods available in resource-constrained health settings of many developing countries. The specificity of the diagnostic method is satisfactory whereas, sensitivity is limited and cannot detect pulmonary tuberculosis (PTB) cases below a bacterial load of 1000 organism/ml. According to the sensitivity of this diagnostic method, PTB patients are categorized into two kinds: (a) smear-positive and (b) smear-negative. Interestingly, this categorization also scales the infectivity of PTB patients in the community. Our current study addresses this heterogeneity in the infectiousness of PTB individuals to investigate its consequences in disease dynamics. A five-dimensional compartmental model is formulated considering the infectivity of both the smear-positive and negative PTB individuals. The expression of the basic reproduction number ($R_0$) is obtained through the next-generation matrix method. The asymptotic behaviour of the model is thoroughly discussed around the steady states of the proposed model. The global asymptotic stability of the equilibrium points is established using suitably constructed Lyapunov functions. It is observed that the disease-free equilibrium is globally asymptotically stable for $R_0<1$ and the disease-persistent equilibrium point is the same whenever $R_0>1$. The study also provides a list of normalized forward sensitivity indices of $R_0$ with respect to the involved parameters. This list showcases the influential level of the associated parameters in determining the size of the threshold quantity $R_0$. It has been found that neglecting the transmission capacity of the smear-negative individuals underestimates the value of $R_0$ whereas, ignoring the smear-negative compartment overestimates the same quantity. We also implement numerical simulations whenever necessary, using a suitable TB parameter set to visualize the obtained analytical results.