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An exact approach to calibrating infectious disease models to surveillance data: The case of HIV and HSV-2
February  2018, 15(1): 181-207. doi: 10.3934/mbe.2018008

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

 1 Istituto di Analisi dei Sistemi ed Informatica 'A. Ruberti' -CNR, Roma, Italy 2 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

Received  September 21, 2016 Accepted  December 06, 2016 Published  May 2017

Fund Project: Federico Papa was supported by SysBioNet, Italian Roadmap Research Infrastructures 2012.

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.

Citation: Federico Papa, Francesca Binda, Giovanni Felici, Marco Franzetti, Alberto Gandolfi, Carmela Sinisgalli, Claudia Balotta. A simple model of HIV epidemic in Italy: The role of the antiretroviral treatment. Mathematical Biosciences & Engineering, 2018, 15 (1) : 181-207. doi: 10.3934/mbe.2018008
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##### References:
Block diagram of the model
Rate of immigration in Italy averaged over each year. Each time label denotes January 1st of the reported year
Per capita loss rate averaged over each year
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
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)
New cases of AIDS and number of deaths by AIDS in Italy (per year). Data from [5], red triangles; model predictions, black circles
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
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
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
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
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
Infection-transfer graph $G$ of model 4. Transfers of individuals between compartments, black arcs; infections, red arcs
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
 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
Predictions for different values of $\beta_5/\beta_2$
 Values at January 1st 2025 ${{\beta }_{5}}/{{\beta }_{2}}$ 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
 Values at January 1st 2025 ${{\beta }_{5}}/{{\beta }_{2}}$ 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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