Mathematical analysis of a two-strain tuberculosis model in Bangladesh
Md Abdul Kuddus, Emma S. McBryde, Adeshina I. Adekunle, Lisa J. White, Michael T. Meehan
Abstract
Abstract Tuberculosis (TB) is an airborne infectious disease that causes millions of deaths worldwide each year (1.2 million people died in 2019). Alarmingly, several strains of the causative agent, Mycobacterium tuberculosis (MTB)—including drug-susceptible (DS) and drug-resistant (DR) variants—already circulate throughout most developing and developed countries, particularly in Bangladesh, with totally drug-resistant strains starting to emerge. In this study we develop a two-strain DS and DR TB transmission model and perform an analysis of the system properties and solutions. Both analytical and numerical results show that the prevalence of drug-resistant infection increases with an increasing drug use through amplification. Both analytic results and numerical simulations suggest that if the basic reproduction numbers of both DS ( $${\text{R}}_{{0{\text{s}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>R</mml:mtext> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>s</mml:mtext> </mml:mrow> </mml:msub> </mml:math> ) and DR ( $${\text{R}}_{{0{\text{r}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>R</mml:mtext> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>r</mml:mtext> </mml:mrow> </mml:msub> </mml:math> ) TB are less than one, i.e. $$\max \left[ {{\text{R}}_{{0{\text{s}}}} ,{\text{ R}}_{{0{\text{r}}}} } \right] < 1,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>max</mml:mo> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mtext>R</mml:mtext> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>s</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>R</mml:mtext> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>r</mml:mtext> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> the disease-free equilibrium is asymptotically stable, meaning that the disease naturally dies out. Furthermore, if $${\text{R}}_{{0{\text{r}}}} > {\text{max}}\left[ {{\text{R}}_{{0{\text{s}}}} ,1} \right]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>R</mml:mtext> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>r</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>></mml:mo> <mml:mtext>max</mml:mtext> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mtext>R</mml:mtext> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>s</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> , then DS TB dies out but DR TB persists in the population, and if $${\text{R}}_{{0{\text{s}}}} > {\text{max}}\left[ {{\text{R}}_{{0{\text{r}}}} ,1} \right]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>R</mml:mtext> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>s</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>></mml:mo> <mml:mtext>max</mml:mtext> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mtext>R</mml:mtext> <mml:mrow> <mml:mn>0</mml:mn> <mml:mtext>r</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> both DS TB and DR TB persist in the population. Further, sensitivity analysis of the model parameters found that the transmission rate of both strains had the greatest influence on DS and DR TB prevalence. We also investigated the effect of treatment rates and amplification on both DS and DR TB prevalence; results indicate that inadequate or inappropriate treatment makes co-existence more likely and increases the relative abundance of DR TB infections.