Mathematical Study on Human Cells Interaction Dynamics for HIV-TB Co-infection

Abstract

Co-infection of Tuberculosis (caused by Mycobacterium tuberculosis bacteria) and HIV (caused by Human Immunodeficiency virus) remains a global burden on public health system and poses particular diagnostic and therapeutic challenges. Due to co-infection, HIV speeds up the progression from latent to active TB and TB bacteria also accelerates the progress of HIV infection which ultimately leads to serious condition in individuals. In this research work, we formulate a six compartment mathematical model on the HIV-TB co-infection dynamics incorporating Macrophage (active and infected), T-cell (active and infected), Virus, and Bacteria population. Moreover, we explore the accelerating effect of both pathogens on each other in mathematical perceptive. Our analytical study reveals the conditions for the persistence of co-infection and also validates the stability criteria of equilibrium points for the disease. We also evaluate the disease-free condition using next generation method, expressed by the basic reproductive ratio ($R_0$). Moreover, our analytical and numerical simulations manifest the influence of certain key parameters on the threats posed by the impact of HIV-TB co-infection.

17.6 Appendix

$A = -\sum a_{ii}$

$B= \sum a_{ii}a_{jj}-\sum a_{ij}a_{ji}$

$C= \sum a_{ij}a_{ji}a_{kk} -\sum a_{ii}a_{jj}a_{kk}-\sum a_{ij}a_{jk}a_{ki}$

$D=\sum a_{ii}a_{jj}a_{kk}a_{ll}+\sum a_{ij}a_{ji}a_{ki}a_{ll}{+}\sum a_{ij}a_{ji}a_{kl}a_{lk}{-}\sum a_{ij}a_{ji}a_{kk}a_{ll}{-}\sum a_{ij}a_{jk}a_{kl}a_{li}$

$E= \sum a_{ij}a_{ji}a_{kk}a_{ll}a_{mm}-\sum a_{ii}a_{jj}a_{kk}a_{ll}a_{mm}-\sum a_{ij}a_{jk}a_{kl}a_{li}a_{mm}-\sum a_{ij}a_{jk}a_{ki}a_{ll}a_{mm}-\sum a_{ij}a_{ji}a_{kl}a_{lk}a_{mm} +\sum a_{ij}a_{jk}a_{kl}a_{lm}a_{mi}$

$F= \sum a_{ii}a_{jj}a_{kk}a_{ll}a_{mm}a_{nn}{+}\sum a_{ij}a_{jk}a_{kl}a_{lm}a_{mi}a_{nn}{+}\sum a_{ij}a_{ji}a_{kl}a_{lk}a_{mm}a_{nn}{+}\sum a_{ij}a_{jk}a_{ki}a_{lm}a_{mn}a_{nl}$ $-\sum a_{ij}a_{ji}a_{kk}a_{ll}a_{mm}a_{nn}-\sum a_{ij}a_{ji}a_{kl}a_{lm}a_{mk}a_{nn}$

Here, A, B, C, D, E, and F follows a rule: $i\ne j \ne k\ne l\ne m \ne n$. In $a_{ij}$, if $i=1$, then j can go to 6 or 5. Following same rule for k, l, m, n. Similarly, 2 can go 5 or 6. 3 can go 5. 4 can go 5 or 3. 5 can go 2 or 3 or 4 or 6 and 6 can go 1 or 2 or 5. In $a_{ij}a_{ji}$, let $i=1$ then j can go 5 or 6. But, $a_{15}a_{51}$ does not exist because 5 cannot go to 1.

Let $B= \sum a_{ii}a_{jj}-\sum a_{ij}a_{ji}$

By above rule $B=\sum _{i=1}^5a_{ii}\sum _{j=i+1}^6 a_{jj}-\{a_{16}a_{61}+a_{26}a_{62}+a_{25}a_{52}+a_{35}a_{53}+a_{45}a_{54}+a_{56}a_{65}\}$.