A Multi-compartment Mathematical Model for HIV–AIDS Transmission and Dynamics

Abstract

In this study, the aim is to formulate a multi-compartment mathematical model regarding the transmission and dynamics of HIV–AIDS. The model is formulated on the basis of a system of linear, ordinary differential equations and admits two locally and globally stable equilibria. Primarily, the existence of the solution of the model and its uniqueness are demonstrated which is then obtained analytically using the fundamental matrix method and eigenvalue approach. The obtained solution serves as the pedestal for studying the dynamics and spread of HIV–AIDS in India. Nevertheless, as an endorsement to the obtained results the simulations are also carried out with model outcomes being contrasted to the exact data of the disease in India.

Appendix

Consider the following system of n non homogeneous linear differential equations in the matrix form

$$\begin{aligned} \dfrac{\text {d}}{\text {d}t} X(t)=AX(t)+B(t) \end{aligned}$$

such that X(t) is an $n\times 1$ column matrix (to be determined), B(t) is also an $n\times 1$ column matrix and A is the coefficient square matrix of order n. If $\lambda _{1}$, $\lambda _{2}$,...,$\lambda _{n}$ denote the eigenvalues of A and $v_{1}$, $v_{2}$,...,$v_{n}$ represent the respective corresponding eigenvectors, then the square matrix F(t) of order n, given as

$$\begin{aligned} F(t)=\left[ v_{1}e^{\lambda _{1}t} ~ v_{2}e^{\lambda _{2}t} ~. ~.~.~ v_{n}e^{\lambda _{n}t} \right] \end{aligned}$$

is called the fundamental matrix corresponding to the homogeneous part $\dfrac{\text {d}}{\text {d}t} X(t)=AX(t)$. Clearly, the matrix F(t) is invertible for all values of t (Ross 2014).