Mathematical Modelling of the Potential Role of Supplementary Feeding for People Living with HIV/AIDS

Abstract

Nutrition plays a critical role in care, support and treatment of HIV infected individuals. In this paper, an HIV/AIDS mathematical model that incorporates supplementary feeding is formulated and analysed using the system of ordinary differential equations. The model has two equilibria, the disease-free equilibrium and endemic equilibrium. Analytical analysis of the model shows that the disease-free and endemic equilibria are globally stable for $R_0<1$ and $R_0>1$ respectively. Numerical simulations are done to show the role of some key model parameters as well as to verify some analytical results. The results of the study show that supplementary feeding taken concurrently with antiretroviral treatment can control the HIV/AIDS transmission by reducing the reproduction number thus leading to the disease control.

Appendix: Analysis of the Reproduction Number

Recall that

$$\begin{aligned}&\rho (FV^{-1})=R_o=R_I+R_{T_n}+R_{T_s}+R_A,\\&R_I=\frac{\beta }{M_1},~ R_{T_n}=\frac{q_1 \eta _1\beta }{M_1M_2 \left( 1-\frac{\gamma q_2}{M_2M_3}\right) },~ R_{T_s}=\frac{q_1q_2\eta _2\beta }{M_1M_2M_3 \left( 1-\frac{\gamma q_2}{M_2M_3}\right) } \end{aligned}$$

and

$$\begin{aligned} R_A=\frac{\left[ M_2M_3\sigma _1+M_3q_1\sigma _2 +q_2q_1\sigma _3\left( 1-\frac{\gamma \sigma _1}{q_1\sigma _3}\right) \right] \eta _3\beta }{M_1M_2M_3M_4 \left( 1-\frac{\gamma q_2}{M_2M_3}\right) }. \end{aligned}$$

A back substitution gives

$$\begin{aligned}&\frac{\beta }{M_1}=\frac{\beta }{\mu +q_1+\sigma _1},\nonumber \\&\frac{M_3q_1\beta \eta _1}{M_1(M_2M_3-\gamma q_2)}=\frac{\eta _1 \beta }{\mu +q_2+\sigma _2}\times \frac{q_1}{\mu +q_1+\sigma _1}\nonumber \\&\qquad \, \times \frac{1}{\left[ 1-\frac{q_2\gamma }{(\mu +q_2+\sigma _2)(\mu +\gamma +\sigma _3)}\right] },\nonumber \\&\frac{q_1q_2\beta \eta _2}{M_1(M_2M_3-\gamma q_2)}=\frac{\eta _2 \beta }{\mu +\gamma +\sigma _3}\times \frac{q_1}{\mu +q_1+\sigma _1}\nonumber \\&\qquad \,\times \frac{q_2}{\mu +q_2+\sigma _2}\times \frac{1}{\left[ 1-\frac{q_2\gamma }{(\mu +q_2+\sigma _2)(\mu +\gamma +\sigma _3)}\right] },\nonumber \\&\frac{[M_2M_3\sigma _1+M_3q_1\sigma _2+q_2(-\gamma \sigma _1+q_1\sigma _3)]\beta \eta _3}{M_1M_4(M_2M_3-\gamma q_2)}\nonumber \\&\quad =\frac{\eta _3 \beta }{\mu +\delta } \left\{ \frac{\sigma _1}{\mu +q_1+\sigma _1}+\left( \frac{q_1}{\mu +q_1+\sigma _1}\times \frac{\sigma _2}{\mu +q_2+\sigma _2}\right) \right. \nonumber \\&\qquad \left. +\,\left( \frac{q_1}{\mu +q_1+\sigma _1}\times \frac{q_2}{\mu +q_2+\sigma _2}\times \frac{\sigma _3}{\mu +\gamma +\sigma _3}\times \left[ 1- \frac{\gamma \sigma _1}{q_1\sigma _3}\right] \right) \right\} \nonumber \\&\qquad \, \times \frac{1}{\left[ 1-\frac{q_2\gamma }{(\mu +q_2+\sigma _2)(\mu +\gamma +\sigma _3)}\right] }. \end{aligned}$$

(A1)

The terms in (A1) can be explained as follows

• $\frac{1}{\mu +q_1+\sigma _1}, \frac{1}{\mu +q_2+\sigma _2}, \frac{1}{\mu +\gamma +\sigma _3}$ and $\frac{1}{\mu +\delta }$ are the average times an individual spends in the infected class I,  the treated class $T_n,$ the supplementary feeding class $T_s$ and the AIDS class A respectively.

• $\frac{q_1}{\mu +q_1+\sigma _1}$ is the proportion of individuals who seek treatment and progress from compartment I to compartment $T_n.$

• $\frac{q_2}{\mu +q_2+\sigma _2}$ is the proportion of individuals who progress from compartment $T_n$ to compartment $T_s.$

• $\frac{\sigma _1}{\mu +q_1+\sigma _1}, \frac{\sigma _2}{\mu +q_2+\sigma _2}$ and $\frac{\sigma _3}{\mu +\gamma +\sigma _3}$ are the proportions of individuals who develop AIDS from compartments I,  $T_n$ and $T_s$ respectively.

• $\frac{q_1}{\mu +q_1+\sigma _1}\times \frac{\sigma _2}{\mu +q_2+\sigma _2}$ is the proportion of individuals who develop AIDS from compartment $T_n$ having come from compartment I.

• $\frac{q_1}{\mu +q_1+\sigma _1}\times \frac{q_2}{\mu +q_2+\sigma _2}$ is the proportion of individuals who progress to compartment $T_s$ from compartment $T_n$ having come from compartment I.

• $\frac{q_1}{\mu +q_1+\sigma _1}\times \frac{q_2}{\mu +q_2+\sigma _2}\times \frac{\sigma _3}{\mu +\gamma +\sigma _3}$ is the proportion of individuals who develop AIDS having started in compartment I via compartments $T_n$ and $T_s$.

$R_I$ is the reproduction number due to infectives, I, $R_{T_n}$ is the reproduction number due to treated, $T_n$, $R_{T_s}$ is the reproduction number due to feeding supplement, $T_s$ and $R_A$ is the reproduction number due to AIDS, A.