A simple model of HIV epidemic in Italy: The role of the antiretroviral treatment

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doi:10.3934/mbe.2018008

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2018, Volume 15, Issue 1: 181-207. Doi: 10.3934/mbe.2018008

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A simple model of HIV epidemic in Italy: The role of the antiretroviral treatment

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Istituto di Analisi dei Sistemi ed Informatica 'A. Ruberti' -CNR, Roma, Italy
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Dipartimento di Scienze Biomediche e Cliniche 'L. Sacco', Sezione di Malattie Infettive e Immunopatologia, Università degli Studi di Milano, Milano, Italy

* Corresponding author: Federico Papa
* Corresponding author: Federico Papa

Received: September 20, 2016

Accepted: December 05, 2016

Published: January 2018

Federico Papa was supported by SysBioNet, Italian Roadmap Research Infrastructures 2012

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Abstract

In the present paper we propose a simple time-varying ODE model to describe the evolution of HIV epidemic in Italy. The model considers a single population of susceptibles, without distinction of high-risk groups within the general population, and accounts for the presence of immigration and emigration, modelling their effects on both the general demography and the dynamics of the infected subpopulations. To represent the intra-host disease progression, the untreated infected population is distributed over four compartments in cascade according to the CD4 counts. A further compartment is added to represent infected people under antiretroviral therapy. The per capita exit rate from treatment, due to voluntary interruption or failure of therapy, is assumed variable with time. The values of the model parameters not reported in the literature are assessed by fitting available epidemiological data over the decade $2003 \div 2012$. Predictions until year 2025 are computed, enlightening the impact on the public health of the early initiation of the antiretroviral therapy. The benefits of this change in the treatment eligibility consist in reducing the HIV incidence rate, the rate of new AIDS cases, and the rate of death from AIDS. Analytical results about properties of the model in its time-invariant form are provided, in particular the global stability of the equilibrium points is established either in the absence and in the presence of infected among immigrants.

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Mathematics Subject Classification: Primary: 34D05, 37B25; Secondary: 92B05, 92D30.

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Figure 1. Block diagram of the model

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Figure 2. Rate of immigration in Italy averaged over each year. Each time label denotes January 1st of the reported year

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Figure 3. Per capita loss rate averaged over each year

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Figure 4. Evolution of the Italian population in the $20 \div 70$ years age range: data (number of inhabitants at the beginning of the indicated year), circle; prediction by Equation 3, solid line

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Figure 5. Time-course of the number of HIV infected individuals and of HAART treated patients in Italy, over the years $2003 \div 2013$. Median values estimated by Camoni et al. [7], circles (bars represent the difference between the 3rd and the 1st quartiles); measurement of the number of treated patients at the end of 2012 [6], square. Model predictions: infected, black solid line; treated patients, black dashed line. Estimate of patients under treatment at year 2005, triangle (communicated by C. Balotta)

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Figure 6. New cases of AIDS and number of deaths by AIDS in Italy (per year). Data from [5], red triangles; model predictions, black circles

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Figure 7. Predictions of new cases of AIDS and number of AIDS deaths (per year). Parameters $\delta_2$, $\delta_3$, $\delta_4$ as in Table 1 (reference prediction), black circles; $\delta_2$, $\delta_4$ unchanged and $\delta_3 = \delta_4$, magenta squares; $\delta_4$ unchanged and $\delta_2 = \delta_3 = \delta_4$, cyan circles. Data from [5], red triangles

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Figure 8. Predictions of the number of infected individuals: total infected, solid lines; infected under therapy, dashed lines. Reference prediction, black; $\delta_3=\delta_4$, magenta; $\delta_2=\delta_3=\delta_4$, cyan. Note that solid lines are substantially overlapping. Red data markers as in Figure 5

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Figure 9. Predictions of the incidence rate (persons$\cdot$day$^{-1}$). Reference prediction, black; $\delta_3=\delta_4$, magenta; $\delta_2=\delta_3=\delta_4$, cyan. Prediction with $\delta_2=\delta_3=\delta_4$ and $\beta_5/\beta_2=0.1$, cyan dashed line

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Figure 10. Predictions of the number of infected individuals: total infected, solid lines; infected under therapy, dashed lines. Reference prediction, black; $1/\xi(t)$ linearly increasing, magenta; $1/\xi(t)$ linearly decreasing, cyan. Note that solid lines are substantially overlapping. Red data markers as in Figure 5

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Figure 11. Predictions of new cases of AIDS and number of AIDS deaths (per year). $\delta_2, \delta_3, \delta_4$ as in Table 1 (reference prediction), black circles; $1/\xi(t)$ linearly increasing, magenta squares; $1/\xi(t)$ linearly decreasing, cyan circles. Data from [5], red triangles

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Figure 12. Infection-transfer graph $G$ of model 4. Transfers of individuals between compartments, black arcs; infections, red arcs

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Table 1. Baseline parameter values

Parameters	Value	Source

$\bar{\Lambda}$	$-156.36$ persons$\cdot$day$^{-1}$	[31]
$\Phi(t=2013)$	$672.70$ persons$\cdot$day$^{-1}$	[31]
$\bar{\mu}$	$1.1 \cdot 10^{-5}$ day$^{-1}$	[31]
$\alpha$	$3.2 \cdot 10^{-3}$	[7]
$1/\xi(t=2003)$	$5,475$ days	Assumed
$1/\xi(t=2013)$	$10,950$ days	Assumed
$\theta_{1}$	$2.86 \cdot 10^{-2}$ day$^{-1}$	[34]
$\theta_{2}$	$4.57 \cdot 10^{-4}$ day$^{-1}$	[34]
$\theta_{3}$	$7.83 \cdot 10^{-4}$ day$^{-1}$	[34,45]
$\theta_{4}$	$1.8 \cdot 10^{-3}$ day$^{-1}$	[34,45]
$\beta_{2}$	$3.17 \cdot 10^{-12}$ (persons$\cdot$day)$^{-1}$	Estimated
$\beta_{1}/\beta_{2}$	$4.5$	[34]
$\beta_{3}/\beta_{2}$	$1.125$	[34]
$\beta_{4}/\beta_{2}$	$1.667$	[34]
$\beta_{5}/\beta_{2}$	$0.2$	[34,6]
δ₂	1.10·10^-19 day^-1	Estimated
δ₃	2.27·10^-3 day^-1	Estimated
δ₄	3.2·10^-3 day^-1	Estimated

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Table 2. Predictions for different values of $\beta_5/\beta_2$

Values at January 1st 2025	$ {{\beta }_{5}}/{{\beta }_{2}} $
Values at January 1st 2025	0.1	0.2	0.3
Infected (persons)	$1.303 \cdot 10^{5}$	$1.368 \cdot 10^{5}$	$1.435 \cdot 10^{5}$
Treated (persons)	$1.117 \cdot 10^{5}$	$1.144 \cdot 10^{5}$	$1.172 \cdot 10^{5}$
HIV infection rate (persons$\cdot$day$^{-1}$)	3.85	5.794	7.816
New cases of AIDS (persons$\cdot$year$^{-1}$)	975.3	1062	1149
AIDS deaths (persons$\cdot$year$^{-1}$)	557.3	606.7	656.8

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