Mathematical model on HIV and nutrition

Figures & data

Table 1. Total protein, albumin, and globulin levels in individuals with and without HIV infection.

	HIV Positive		HIV Negative
	Men	Women	Men	Women	Reference
Total protein g/L	$76.0\pm 1.3$	$79.3\pm 1.5$	$72.3\pm 0.1$	$71.2\pm 0.1$
Albumin g/L	$41.7\pm 0.6$	$38.6\pm 0.8$	$44.6\pm 0.1$	$42.0\pm 0.1$	[44]
Total Protein g/dL	$7.72(7.45-8)$		$7.8(7.57-8.02)$
Albumin g/dL	$4.4(4.27-4.54)$		$4.59(4.25-4.93)$		[46]
IgG g/dL	N/A		N/A
Total protein g/L, mean (SD)	$85.4\left(7.9\right)$		$75.5\left(6.1\right)$
Albumin g/L, mean (SD)	$40\left(3.7\right)$		$43\left(4.3\right)$		[47]
IgG g/L, mean (SD)	$27\left(8.7\right)$		$16\left(3.9\right)$
IgG mg/mL	1998 (655–4411)		1458 (710–3122)
IgA mg/dL	294 (47–1200)		232 (110–593)		[48]
IgM mg/dL	206 (62–661)		284 (47–709)

Figure 1. Schematic diagram of the within-host model. CD4 T cells are denoted by T, infected cells are denoted by $T_i$, the virus is denoted by V , CD8 T cells are denoted by Z and nutrition (protein) is denoted by η.

Table 2. Brief descriptions of the parameters of the within-host model together with the numerical values and units of the parameters and initial conditions.

Param.	Description	Value	Units
$T\left(0\right)$	Initial value for target cells	10	cells/μL
$T_i\left(0\right)$	Initial value for infected cells	0	cells/μL
$V\left(0\right)$	Initial value for the virus	${10}^{-9}$	virions/μL
$Z\left(0\right)$	Initial value for immune cells	6	cells/μL
$\eta \left(0\right)$	Initial value for total protein	5	gr
$r_0$	Target cell production	10	1/μL.day
r	Target cell production per gram total protein	10/350	1/μL.day.gr
β	Infection rate of target cells	$6.5\times {10}^{-4}$	μL/virion.day
${\rho }_0$	Infection rate of target cells times gram	$6.5\times {10}^{-4}\times 350$	μL.gr/virion.day
Ψ	Half saturation constant for nutrition	175	gr
A	Proportionality constant for nutrition	0.1	1/gr
d	Death rate of uninfected cells	0.01	1/ day
δ	Death rate of infected cells	1	1/day
p	Virus production rate	2000	virions/cell.day
c	Clearance rate of virus	23	1/day
${\lambda }_0$	CD8 T cell recruitment	1	cell/μL.day
${\lambda }_z$	CD8 T cell recruitment per gram total protein	${10}^{-3}/350$	cells/μL.day.gr
${\psi }_0$	Strength of CD8 T response	0.42	$\text{μ}\mathrm{L}$/cell.day
${\mu }_z$	Death rate of CD8 T cells	2	1/day
${\lambda }_{\eta }$	Rate of ingestion of total protein	65	gr/day
${\text{μ}}_{\eta }$	Clearance of total protein in the body	1.1857	1/day
${\gamma }_{\eta }$	Enhancement rate of protein by virus	0.0001	1/day
b	Antigen driven activation rate of CD8 T	1	1/day
${\mu }_V$	Viral clearance by immunoglobulins	0.1057	1/gr.day

Figure 2. Graphical representation of inequality (14(14) $\begin{array}{rl}& p{\mu }_Z({\mu }_{\eta }-{\gamma }_{\eta }{V}^{\ast })>b{V}^{\ast }\left[c\right({\mu }_{\eta }-{\gamma }_{\eta }{V}^{\ast })+{\lambda }_{\eta }{\mu }_V]\\ & \iff b{\lambda }_{\eta }{\mu }_V{V}^{\ast }<({\mu }_{\eta }-{\gamma }_{\eta }{V}^{\ast })(p{\mu }_Z-bc{V}^{\ast }).\end{array}$(14) ).

Figure 3. The impact of the viral production rate (p) on (a) the viral load and (b) total protein. (a) Simulations of the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}T}{\mathrm{d}\tau }& =r\eta -\rho \left(\eta \right)TV-\mathrm{d}T,\\ \frac{\mathrm{d}{T}_i}{\mathrm{d}\tau }& =\rho \left(\eta \right)TV-\delta T_i-\psi \left(\eta \right)T_{i}Z,\\ \frac{\mathrm{d}V}{\mathrm{d}\tau }& =p{T}_i-cV-{\mu }_V\eta V,\\ \frac{\mathrm{d}Z}{\mathrm{d}\tau }& ={\lambda }_z\eta +b{T}_{i}Z-{\mu }_{z}Z,\\ \frac{\mathrm{d}\eta }{\mathrm{d}\tau }& ={\lambda }_{\eta }-{\mu }_{\eta }\eta +{\gamma }_{\eta }\eta V.\end{array}$(1) ) depicting the impact of the viral production rate (p) on the viral load. The other parameters used for the simulations are presented in Table 2. (b) Simulations of the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}T}{\mathrm{d}\tau }& =r\eta -\rho \left(\eta \right)TV-\mathrm{d}T,\\ \frac{\mathrm{d}{T}_i}{\mathrm{d}\tau }& =\rho \left(\eta \right)TV-\delta T_i-\psi \left(\eta \right)T_{i}Z,\\ \frac{\mathrm{d}V}{\mathrm{d}\tau }& =p{T}_i-cV-{\mu }_V\eta V,\\ \frac{\mathrm{d}Z}{\mathrm{d}\tau }& ={\lambda }_z\eta +b{T}_{i}Z-{\mu }_{z}Z,\\ \frac{\mathrm{d}\eta }{\mathrm{d}\tau }& ={\lambda }_{\eta }-{\mu }_{\eta }\eta +{\gamma }_{\eta }\eta V.\end{array}$(1) ) depicting the impact of the viral production rate (p) on total protein. The other parameters used for the simulations are presented in Table 2.

Figure 4. The impact of the protein intake (${\lambda }_{\eta }$) on (a) the viral load and (b) total protein. (a) Simulations of the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}T}{\mathrm{d}\tau }& =r\eta -\rho \left(\eta \right)TV-\mathrm{d}T,\\ \frac{\mathrm{d}{T}_i}{\mathrm{d}\tau }& =\rho \left(\eta \right)TV-\delta T_i-\psi \left(\eta \right)T_{i}Z,\\ \frac{\mathrm{d}V}{\mathrm{d}\tau }& =p{T}_i-cV-{\mu }_V\eta V,\\ \frac{\mathrm{d}Z}{\mathrm{d}\tau }& ={\lambda }_z\eta +b{T}_{i}Z-{\mu }_{z}Z,\\ \frac{\mathrm{d}\eta }{\mathrm{d}\tau }& ={\lambda }_{\eta }-{\mu }_{\eta }\eta +{\gamma }_{\eta }\eta V.\end{array}$(1) ) depicting the impact of the protein intake (${\lambda }_{\eta }$) on the viral load. The other parameters used for the simulations are presented in Table 2. (b) Simulations of the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}T}{\mathrm{d}\tau }& =r\eta -\rho \left(\eta \right)TV-\mathrm{d}T,\\ \frac{\mathrm{d}{T}_i}{\mathrm{d}\tau }& =\rho \left(\eta \right)TV-\delta T_i-\psi \left(\eta \right)T_{i}Z,\\ \frac{\mathrm{d}V}{\mathrm{d}\tau }& =p{T}_i-cV-{\mu }_V\eta V,\\ \frac{\mathrm{d}Z}{\mathrm{d}\tau }& ={\lambda }_z\eta +b{T}_{i}Z-{\mu }_{z}Z,\\ \frac{\mathrm{d}\eta }{\mathrm{d}\tau }& ={\lambda }_{\eta }-{\mu }_{\eta }\eta +{\gamma }_{\eta }\eta V.\end{array}$(1) ) depicting the impact of the protein intake (${\lambda }_{\eta }$) on total protein. The other parameters used for the simulations are presented in Table 2.

Figure 5. (a) The impact of the protein intake (${\lambda }_{\eta }$) on the viral load and (b) the dynamics of the viral load (V) and total protein (η). (a) Simulations of the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}T}{\mathrm{d}\tau }& =r\eta -\rho \left(\eta \right)TV-\mathrm{d}T,\\ \frac{\mathrm{d}{T}_i}{\mathrm{d}\tau }& =\rho \left(\eta \right)TV-\delta T_i-\psi \left(\eta \right)T_{i}Z,\\ \frac{\mathrm{d}V}{\mathrm{d}\tau }& =p{T}_i-cV-{\mu }_V\eta V,\\ \frac{\mathrm{d}Z}{\mathrm{d}\tau }& ={\lambda }_z\eta +b{T}_{i}Z-{\mu }_{z}Z,\\ \frac{\mathrm{d}\eta }{\mathrm{d}\tau }& ={\lambda }_{\eta }-{\mu }_{\eta }\eta +{\gamma }_{\eta }\eta V.\end{array}$(1) ) illustrating the impact of the protein intake rate (${\lambda }_{\eta }$) on viral load. For these simulations, $p=6000,{\psi }_0=200$, and the other parameter values are as given in Table 2. (b) Simulations of the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}T}{\mathrm{d}\tau }& =r\eta -\rho \left(\eta \right)TV-\mathrm{d}T,\\ \frac{\mathrm{d}{T}_i}{\mathrm{d}\tau }& =\rho \left(\eta \right)TV-\delta T_i-\psi \left(\eta \right)T_{i}Z,\\ \frac{\mathrm{d}V}{\mathrm{d}\tau }& =p{T}_i-cV-{\mu }_V\eta V,\\ \frac{\mathrm{d}Z}{\mathrm{d}\tau }& ={\lambda }_z\eta +b{T}_{i}Z-{\mu }_{z}Z,\\ \frac{\mathrm{d}\eta }{\mathrm{d}\tau }& ={\lambda }_{\eta }-{\mu }_{\eta }\eta +{\gamma }_{\eta }\eta V.\end{array}$(1) ) illustrating the dynamics of the viral load (V) and total protein (η). Parameter values are as fixed at their baseline values given in Table 2.

Figure 6. The impact of the protein intake (${\gamma }_{\eta }$) on (a) the viral load and (b) total protein. (a) Time series plot of the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}T}{\mathrm{d}\tau }& =r\eta -\rho \left(\eta \right)TV-\mathrm{d}T,\\ \frac{\mathrm{d}{T}_i}{\mathrm{d}\tau }& =\rho \left(\eta \right)TV-\delta T_i-\psi \left(\eta \right)T_{i}Z,\\ \frac{\mathrm{d}V}{\mathrm{d}\tau }& =p{T}_i-cV-{\mu }_V\eta V,\\ \frac{\mathrm{d}Z}{\mathrm{d}\tau }& ={\lambda }_z\eta +b{T}_{i}Z-{\mu }_{z}Z,\\ \frac{\mathrm{d}\eta }{\mathrm{d}\tau }& ={\lambda }_{\eta }-{\mu }_{\eta }\eta +{\gamma }_{\eta }\eta V.\end{array}$(1) ) showing the effects of the enhancement rate of protein by the virus (${\gamma }_{\eta }$) on the viral load (V). The values of the other parameter used for the simulations are as given in Table 2. (b) Time series plot of the model (1(1) $\begin{array}{rl}\frac{\mathrm{d}T}{\mathrm{d}\tau }& =r\eta -\rho \left(\eta \right)TV-\mathrm{d}T,\\ \frac{\mathrm{d}{T}_i}{\mathrm{d}\tau }& =\rho \left(\eta \right)TV-\delta T_i-\psi \left(\eta \right)T_{i}Z,\\ \frac{\mathrm{d}V}{\mathrm{d}\tau }& =p{T}_i-cV-{\mu }_V\eta V,\\ \frac{\mathrm{d}Z}{\mathrm{d}\tau }& ={\lambda }_z\eta +b{T}_{i}Z-{\mu }_{z}Z,\\ \frac{\mathrm{d}\eta }{\mathrm{d}\tau }& ={\lambda }_{\eta }-{\mu }_{\eta }\eta +{\gamma }_{\eta }\eta V.\end{array}$(1) ) showing the effects of the enhancement rate of protein by the virus (${\gamma }_{\eta }$) on the total protein level (η). The values of the other parameter used for the simulations are as given in Table 2.

SV Thuppal, S Jun, A Cowan, et al. The nutritional status of HIV-infected US adults. Curr Dev Nutr. 2017;1:Article ID e001636. doi: 10.3945/cdn.117.001636

 PubMedGoogle Scholar

Z Nozarian, V Mehrtash, A Abdollahi, et al. Serum protein electrophoresis pattern in patients living with HIV: frequency of possible abnormalities in Iranian patients. Iran J Microbiol. 2019;11:440.

 PubMed Web of Science ®Google Scholar

AE Zemlin, H Ipp, S Maleka, et al. Serum protein electrophoresis patterns in human immunodeficiency virus-infected individuals not on antiretroviral treatment. Ann Clin Biochem. 2015;52:346–351. doi: 10.1177/0004563214565824

 PubMed Web of Science ®Google Scholar

JP McGowan, SS Shah, CB Small, et al. Relationship of serum immunoglobulin and IgG subclass levels to race, ethnicity and behavioral characteristics in HIV infection. Med Sci Monit. 2006;12:CR11–CR16.

 PubMed Web of Science ®Google Scholar