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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
References:
[1]

M. ArcàC. A. Perucci and T. Spadea, The epidemic dynamics of HIV-1 in Italy: Modelling the interaction between intravenous drug users and heterosexual population, Stat Med, 11 (1992), 1657-1684. Google Scholar

[2]

S. M. BlowerH. B. Gershengorn and R. M. Grant, A tale of two futures: HIV and antiretroviral therapy in San Francisco, Science, 287 (2000), 650-654. doi: 10.1126/science.287.5453.650. Google Scholar

[3]

C. BonifaziF. HeinsS. Strozza and M. Vitiello, The italian transition from emigration to immigration country, IRPPS Working Papers, 24 (2009), 2-91. Google Scholar

[4]

L. CamoniS. BorosV. RegineM. Ferri and M. Santaquilani, Aggiornamento delle nuove diagnosi di infezione da HIV e dei casi di AIDS in Italia al 31 dicembre 2012, Notiziario Istituto Superiore di Sanitá, 26 (2013), 3-47. Google Scholar

[5]

L. CamoniS. BorosV. RegineM. Santaquilani and M. Ferri, Aggiornamento delle nuove diagnosi di infezione da HIV e dei casi di AIDS in Italia al 31 dicembre 2013, Notiziario Istituto Superiore di Sanitá, 27 (2014), 3-46. Google Scholar

[6]

L. CamoniM. RaimondoM. DorrucciV. Regine and M. C. Salfa, Estimating minimum adult HIV prevalence: A cross-sectional study to assess the characteristics of people living with HIV in Italy, AIDS Res Hum Retrov, 31 (2015), 282-287. doi: 10.1089/aid.2014.0154. Google Scholar

[7]

L. Camoni, V. Regine, K. Stanecki, M. C. Salfa and M. Raimondo et al. , Estimates of the number of people living with HIV in Italy, Biomed Res Int, 2014 (2014), Article ID 209619, 6 pages. doi: 10.1155/2014/209619. Google Scholar

[8]

N. CrepazT. A. Hart and G. Marks, Highly active antiretroviral therapy and sexual risk behavior: A meta-analytic review, JAMA-J Am Med Assoc, 292 (2004), 224-236. doi: 10.1001/jama.292.2.224. Google Scholar

[9]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J Math Biol, 28 (1990), 365-382. doi: 10.1007/BF00178324. Google Scholar

[10]

D. DonnellJ. M. BaetenJ. KiarieK. K. Thomas and W. Stevens, Heterosexual HIV-1 transmission after initiation of antiretroviral therapy: A prospective cohort analysis, Lancet, 375 (2010), 2092-2098. doi: 10.1016/S0140-6736(10)60705-2. Google Scholar

[11]

J. EatonN. MenziesJ. StoverV. Cambiano and L. Chindelevitch, Health benefits, costs, and cost-effectiveness of earlier eligibility for adult antiretroviral therapy and expanded treatment coverage: a combined analysis of 12 mathematical models, Lancet Glob Health, 2 (2014), e23-e34. doi: 10.1016/S2214-109X(13)70172-4. Google Scholar

[12]

M. FranzettiF. AdorniC. ParraviciniB. Vergani and S. Antinori, Trends and predictors of non-AIDS-defining cancers in men and women with HIV infection: A single-institution retrospective study before and after the introduction of HAART, J Acquir Immune Defic Syndr, 62 (2013), 414-420. doi: 10.1097/QAI.0b013e318282a189. Google Scholar

[13]

M. Franzetti, M. Violin, A. Antinori, A. De Luca and F. Ceccherini-Silberstein et al. , Trends and correlates of HIV-1 resistance among subjects failing an antiretroviral treatment over the 2003-2012 decade in Italy, BMC Infect Dis, 14 (2014), p398. doi: 10.1186/1471-2334-14-398. Google Scholar

[14]

R. Granich, S. Crowley, M. Vitoria, Y. R. Lo and Y. Souteyrand et al. , Highly active antiretroviral treatment for the prevention of HIV transmission, J Int AIDS Soc, 13 (2010), 1pp.Google Scholar

[15]

R. M. GranichC. F. GilksC. DyeK. M. De Cock and B. G. Williams, Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission: A mathematical model, Lancet, 373 (2009), 48-57. doi: 10.1016/S0140-6736(08)61697-9. Google Scholar

[16]

H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discret Contin Dyn S -Series B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. Google Scholar

[17]

H. GuoM. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad Appl Math Quart, 14 (2006), 259-284. Google Scholar

[18]

H. GuoM. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J Appl Math, 72 (2012), 261-279. doi: 10.1137/110827028. Google Scholar

[19]

M. IannelliF. MilnerA. Pugliese and M. Gonzo, The HIV/AIDS epidemics among drug injectors: A study of contact structure through a mathematical model, Math Biosci, 139 (1997), 25-58. doi: 10.1016/S0025-5564(96)00137-X. Google Scholar

[20]

S. C. KalichmanL. EatonD. CainC. Cherry and H. Pope, HIV treatment beliefs and sexual transmission risk behaviors among HIV positive men and women, J Behav Med, 29 (2006), 401-410. doi: 10.1007/s10865-006-9066-3. Google Scholar

[21]

M. KretzschmarM. Schim van der LoeffP. BirrellD. De Angelis and R. Coutinho, Prospects of elimination of HIV with test-and-treat strategy, Proc Natl Acad Sci USA, 110 (2013), 15538-15543. doi: 10.1073/pnas.1301801110. Google Scholar

[22]

J. R. Lingappa, J. P. Hughes, R. S. Wang, J. M. Baeten and C. Celum et al. , Estimating the impact of plasma HIV-1 RNA reductions on heterosexual HIV-1 transmission risk, PLoS One, 5 (2010), e12598. doi: 10.1371/journal.pone.0012598. Google Scholar

[23]

P. M. Luz, B. Grinsztejn, L. Velasque, A. G. Pacheco and V. G. Veloso et al. , Long-term CD4 + cell count in response to combination antiretroviral therapy, PLoS One, 9 (2014), e93039. doi: 10.1371/journal.pone.0093039. Google Scholar

[24]

K. A. LythgoeL. Pellis and C. Fraser, Is HIV short-sighted? Insights from a multistrain nested model, Evolution, 67 (2013), 2769-2782. doi: 10.1111/evo.12166. Google Scholar

[25]

R. ManfrediL. Calza and F. Chiodo, HIV disease among immigrants coming to Italy from outside of the European Union: A case-control study of epidemiological and clinical features, Epidemiol Infect, 127 (2001), 527-533. doi: 10.1017/S0950268801006227. Google Scholar

[26]

R. May and R. Anderson, Transmission dynamics of HIV infection, Nature, 326 (1987), 137-142. doi: 10.1038/326137a0. Google Scholar

[27]

V. MillerC. A. SabinA. N. PhillipsC. Rottmann and H. Rabenau, The impact of protease inhibitor-containing highly active antiretroviral therapy on progression of HIV disease and its relationship to CD4 and viral load, AIDS, 14 (2000), 2129-2136. doi: 10.1097/00002030-200009290-00009. Google Scholar

[28]

F. NyabadzaZ. Mukandavire and S. Hove-Musekwa, Modelling the HIV/AIDS epidemic trends in South Africa: Insights from a simple mathematical model, Nonlinear Anal Real World Appl, 12 (2011), 2091-2104. doi: 10.1016/j.nonrwa.2010.12.024. Google Scholar

[29]

P. Paci, F. Martini, M. Bernaschi, G. D'Offizi and F. Castiglione, Timely HAART initiation may pave the way for a better viral control, BMC Infect Dis, 11 (2011), p56. doi: 10.1186/1471-2334-11-56. Google Scholar

[30]

M. ParczewskiM. Leszczyszyn-PynkaM. Witak-JedraK. Maciejewska and W. Rymer, Transmitted HIV drug resistance in antiretroviral-treatment-naive patients from poland differs by transmission category and subtype, J Antimicrob Chemother, 70 (2015), 233-242. doi: 10.1093/jac/dku372. Google Scholar

[31]

V. Patruno, M. Venturi and S. Roberto, Intercensal Population Estimates. Demographic Balance, National Institute of Statistics of Italy (ISTAT), http://demo.istat.it/index.html.Google Scholar

[32]

T. C. QuinnM. J. WawerN. SewankamboD. Serwadda and C. Li, Viral load and heterosexual transmission of human immunodeficiency virus type 1. Rakai Project Study Group, N Engl J Med, 343 (2000), 921-929. Google Scholar

[33]

A. S. RaoK. ThomasK. Sudhakar and P. K. Maini, HIV/AIDS epidemic in India and predicting the impact of the national response: mathematical modeling and analyis, Math Biosci Eng, 6 (2009), 779-813. doi: 10.3934/mbe.2009.6.779. Google Scholar

[34]

R. J. SmithJ. T. OkanoJ. S. KahnE. N. Bodine and S. Blower, Evolutionary dynamics of complex networks of HIV drug-resistant strains: The case of San Francisco, Science, 327 (2010), 697-701. doi: 10.1126/science.1180556. Google Scholar

[35]

T. SterlingR. E. Chaisson and R. D. Moore, HIV-1 RNA, CD4 T-lymphocytes, and clinical response to highly active antiretroviral therapy, AIDS, 15 (2001), 2251-2257. Google Scholar

[36]

B. TaiwoR. L. Murphy and C. Katlama, Novel antiretroviral combinations in treatment-experienced patients with HIV infection: Rationale and results, Drugs, 70 (2010), 1629-1642. doi: 10.2165/11538020-000000000-00000. Google Scholar

[37]

The INSIGHT START Study GroupJ. D. LundgrenA. G. BabikerF. Gordin and S. Emery, Initiation of antiretroviral therapy in early asymptomatic HIV infection, N Engl J Med, 373 (2015), 795-807. Google Scholar

[38]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[39]

J. VercauterenA. M. WensingD. A. van de VijverJ. Albert and C. Balotta, Transmission of drug-resistant HIV-1 is stabilizing in europe, J Infect Dis, 200 (2009), 1503-1508. doi: 10.1086/644505. Google Scholar

[40]

R. WeberM. RuppikM. RickenbachA. Spoerri and H. Furrer, Decreasing mortality and changing patterns of causes of death in the Swiss HIV Cohort Study, HIV Med, 14 (2013), 195-207. doi: 10.1111/j.1468-1293.2012.01051.x. Google Scholar

[41]

When To Start ConsortiumJ. A. SterneM. MayD. Costagliola and F. de Wolf, Timing of initiation of antiretroviral therapy in AIDS-free HIV-1-infected patients: A collaborative analysis of 18 HIV cohort studies, Lancet, 373 (2009), 1352-1363. Google Scholar

[42]

WHO Guidelines Approved by the Guidelines Review Committee, Guideline on When to Start Antiretroviral Therapy and on Pre-Exposure Prophylaxis for HIV, World Health Organization, Geneva, 2015.Google Scholar

[43]

D. P. WilsonM. G. LawA. E. GrulichD. A. Cooper and J. M. Kaldor, Relation between HIV viral load and infectiousness: A model-based analysis, Lancet, 372 (2008), 314-320. doi: 10.1016/S0140-6736(08)61115-0. Google Scholar

[44]

T. ZhangM. JiaH. LuoY. Zhou and N. Wang, Study on a HIV/AIDS model with application to Yunnan province, China, Appl Math Model, 35 (2011), 4379-4392. doi: 10.1016/j.apm.2011.03.004. Google Scholar

[45]

M. Zwahlen and M. Egger, Progression and Mortality of Untreated HIV-positive Individuals Living in Resource-Limited Settings: Update of Literature Review and Evidence Synthesis, Report to UNAIDS obligation no HQ/05/422204.Google Scholar

show all references

References:
[1]

M. ArcàC. A. Perucci and T. Spadea, The epidemic dynamics of HIV-1 in Italy: Modelling the interaction between intravenous drug users and heterosexual population, Stat Med, 11 (1992), 1657-1684. Google Scholar

[2]

S. M. BlowerH. B. Gershengorn and R. M. Grant, A tale of two futures: HIV and antiretroviral therapy in San Francisco, Science, 287 (2000), 650-654. doi: 10.1126/science.287.5453.650. Google Scholar

[3]

C. BonifaziF. HeinsS. Strozza and M. Vitiello, The italian transition from emigration to immigration country, IRPPS Working Papers, 24 (2009), 2-91. Google Scholar

[4]

L. CamoniS. BorosV. RegineM. Ferri and M. Santaquilani, Aggiornamento delle nuove diagnosi di infezione da HIV e dei casi di AIDS in Italia al 31 dicembre 2012, Notiziario Istituto Superiore di Sanitá, 26 (2013), 3-47. Google Scholar

[5]

L. CamoniS. BorosV. RegineM. Santaquilani and M. Ferri, Aggiornamento delle nuove diagnosi di infezione da HIV e dei casi di AIDS in Italia al 31 dicembre 2013, Notiziario Istituto Superiore di Sanitá, 27 (2014), 3-46. Google Scholar

[6]

L. CamoniM. RaimondoM. DorrucciV. Regine and M. C. Salfa, Estimating minimum adult HIV prevalence: A cross-sectional study to assess the characteristics of people living with HIV in Italy, AIDS Res Hum Retrov, 31 (2015), 282-287. doi: 10.1089/aid.2014.0154. Google Scholar

[7]

L. Camoni, V. Regine, K. Stanecki, M. C. Salfa and M. Raimondo et al. , Estimates of the number of people living with HIV in Italy, Biomed Res Int, 2014 (2014), Article ID 209619, 6 pages. doi: 10.1155/2014/209619. Google Scholar

[8]

N. CrepazT. A. Hart and G. Marks, Highly active antiretroviral therapy and sexual risk behavior: A meta-analytic review, JAMA-J Am Med Assoc, 292 (2004), 224-236. doi: 10.1001/jama.292.2.224. Google Scholar

[9]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J Math Biol, 28 (1990), 365-382. doi: 10.1007/BF00178324. Google Scholar

[10]

D. DonnellJ. M. BaetenJ. KiarieK. K. Thomas and W. Stevens, Heterosexual HIV-1 transmission after initiation of antiretroviral therapy: A prospective cohort analysis, Lancet, 375 (2010), 2092-2098. doi: 10.1016/S0140-6736(10)60705-2. Google Scholar

[11]

J. EatonN. MenziesJ. StoverV. Cambiano and L. Chindelevitch, Health benefits, costs, and cost-effectiveness of earlier eligibility for adult antiretroviral therapy and expanded treatment coverage: a combined analysis of 12 mathematical models, Lancet Glob Health, 2 (2014), e23-e34. doi: 10.1016/S2214-109X(13)70172-4. Google Scholar

[12]

M. FranzettiF. AdorniC. ParraviciniB. Vergani and S. Antinori, Trends and predictors of non-AIDS-defining cancers in men and women with HIV infection: A single-institution retrospective study before and after the introduction of HAART, J Acquir Immune Defic Syndr, 62 (2013), 414-420. doi: 10.1097/QAI.0b013e318282a189. Google Scholar

[13]

M. Franzetti, M. Violin, A. Antinori, A. De Luca and F. Ceccherini-Silberstein et al. , Trends and correlates of HIV-1 resistance among subjects failing an antiretroviral treatment over the 2003-2012 decade in Italy, BMC Infect Dis, 14 (2014), p398. doi: 10.1186/1471-2334-14-398. Google Scholar

[14]

R. Granich, S. Crowley, M. Vitoria, Y. R. Lo and Y. Souteyrand et al. , Highly active antiretroviral treatment for the prevention of HIV transmission, J Int AIDS Soc, 13 (2010), 1pp.Google Scholar

[15]

R. M. GranichC. F. GilksC. DyeK. M. De Cock and B. G. Williams, Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission: A mathematical model, Lancet, 373 (2009), 48-57. doi: 10.1016/S0140-6736(08)61697-9. Google Scholar

[16]

H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discret Contin Dyn S -Series B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. Google Scholar

[17]

H. GuoM. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad Appl Math Quart, 14 (2006), 259-284. Google Scholar

[18]

H. GuoM. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J Appl Math, 72 (2012), 261-279. doi: 10.1137/110827028. Google Scholar

[19]

M. IannelliF. MilnerA. Pugliese and M. Gonzo, The HIV/AIDS epidemics among drug injectors: A study of contact structure through a mathematical model, Math Biosci, 139 (1997), 25-58. doi: 10.1016/S0025-5564(96)00137-X. Google Scholar

[20]

S. C. KalichmanL. EatonD. CainC. Cherry and H. Pope, HIV treatment beliefs and sexual transmission risk behaviors among HIV positive men and women, J Behav Med, 29 (2006), 401-410. doi: 10.1007/s10865-006-9066-3. Google Scholar

[21]

M. KretzschmarM. Schim van der LoeffP. BirrellD. De Angelis and R. Coutinho, Prospects of elimination of HIV with test-and-treat strategy, Proc Natl Acad Sci USA, 110 (2013), 15538-15543. doi: 10.1073/pnas.1301801110. Google Scholar

[22]

J. R. Lingappa, J. P. Hughes, R. S. Wang, J. M. Baeten and C. Celum et al. , Estimating the impact of plasma HIV-1 RNA reductions on heterosexual HIV-1 transmission risk, PLoS One, 5 (2010), e12598. doi: 10.1371/journal.pone.0012598. Google Scholar

[23]

P. M. Luz, B. Grinsztejn, L. Velasque, A. G. Pacheco and V. G. Veloso et al. , Long-term CD4 + cell count in response to combination antiretroviral therapy, PLoS One, 9 (2014), e93039. doi: 10.1371/journal.pone.0093039. Google Scholar

[24]

K. A. LythgoeL. Pellis and C. Fraser, Is HIV short-sighted? Insights from a multistrain nested model, Evolution, 67 (2013), 2769-2782. doi: 10.1111/evo.12166. Google Scholar

[25]

R. ManfrediL. Calza and F. Chiodo, HIV disease among immigrants coming to Italy from outside of the European Union: A case-control study of epidemiological and clinical features, Epidemiol Infect, 127 (2001), 527-533. doi: 10.1017/S0950268801006227. Google Scholar

[26]

R. May and R. Anderson, Transmission dynamics of HIV infection, Nature, 326 (1987), 137-142. doi: 10.1038/326137a0. Google Scholar

[27]

V. MillerC. A. SabinA. N. PhillipsC. Rottmann and H. Rabenau, The impact of protease inhibitor-containing highly active antiretroviral therapy on progression of HIV disease and its relationship to CD4 and viral load, AIDS, 14 (2000), 2129-2136. doi: 10.1097/00002030-200009290-00009. Google Scholar

[28]

F. NyabadzaZ. Mukandavire and S. Hove-Musekwa, Modelling the HIV/AIDS epidemic trends in South Africa: Insights from a simple mathematical model, Nonlinear Anal Real World Appl, 12 (2011), 2091-2104. doi: 10.1016/j.nonrwa.2010.12.024. Google Scholar

[29]

P. Paci, F. Martini, M. Bernaschi, G. D'Offizi and F. Castiglione, Timely HAART initiation may pave the way for a better viral control, BMC Infect Dis, 11 (2011), p56. doi: 10.1186/1471-2334-11-56. Google Scholar

[30]

M. ParczewskiM. Leszczyszyn-PynkaM. Witak-JedraK. Maciejewska and W. Rymer, Transmitted HIV drug resistance in antiretroviral-treatment-naive patients from poland differs by transmission category and subtype, J Antimicrob Chemother, 70 (2015), 233-242. doi: 10.1093/jac/dku372. Google Scholar

[31]

V. Patruno, M. Venturi and S. Roberto, Intercensal Population Estimates. Demographic Balance, National Institute of Statistics of Italy (ISTAT), http://demo.istat.it/index.html.Google Scholar

[32]

T. C. QuinnM. J. WawerN. SewankamboD. Serwadda and C. Li, Viral load and heterosexual transmission of human immunodeficiency virus type 1. Rakai Project Study Group, N Engl J Med, 343 (2000), 921-929. Google Scholar

[33]

A. S. RaoK. ThomasK. Sudhakar and P. K. Maini, HIV/AIDS epidemic in India and predicting the impact of the national response: mathematical modeling and analyis, Math Biosci Eng, 6 (2009), 779-813. doi: 10.3934/mbe.2009.6.779. Google Scholar

[34]

R. J. SmithJ. T. OkanoJ. S. KahnE. N. Bodine and S. Blower, Evolutionary dynamics of complex networks of HIV drug-resistant strains: The case of San Francisco, Science, 327 (2010), 697-701. doi: 10.1126/science.1180556. Google Scholar

[35]

T. SterlingR. E. Chaisson and R. D. Moore, HIV-1 RNA, CD4 T-lymphocytes, and clinical response to highly active antiretroviral therapy, AIDS, 15 (2001), 2251-2257. Google Scholar

[36]

B. TaiwoR. L. Murphy and C. Katlama, Novel antiretroviral combinations in treatment-experienced patients with HIV infection: Rationale and results, Drugs, 70 (2010), 1629-1642. doi: 10.2165/11538020-000000000-00000. Google Scholar

[37]

The INSIGHT START Study GroupJ. D. LundgrenA. G. BabikerF. Gordin and S. Emery, Initiation of antiretroviral therapy in early asymptomatic HIV infection, N Engl J Med, 373 (2015), 795-807. Google Scholar

[38]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[39]

J. VercauterenA. M. WensingD. A. van de VijverJ. Albert and C. Balotta, Transmission of drug-resistant HIV-1 is stabilizing in europe, J Infect Dis, 200 (2009), 1503-1508. doi: 10.1086/644505. Google Scholar

[40]

R. WeberM. RuppikM. RickenbachA. Spoerri and H. Furrer, Decreasing mortality and changing patterns of causes of death in the Swiss HIV Cohort Study, HIV Med, 14 (2013), 195-207. doi: 10.1111/j.1468-1293.2012.01051.x. Google Scholar

[41]

When To Start ConsortiumJ. A. SterneM. MayD. Costagliola and F. de Wolf, Timing of initiation of antiretroviral therapy in AIDS-free HIV-1-infected patients: A collaborative analysis of 18 HIV cohort studies, Lancet, 373 (2009), 1352-1363. Google Scholar

[42]

WHO Guidelines Approved by the Guidelines Review Committee, Guideline on When to Start Antiretroviral Therapy and on Pre-Exposure Prophylaxis for HIV, World Health Organization, Geneva, 2015.Google Scholar

[43]

D. P. WilsonM. G. LawA. E. GrulichD. A. Cooper and J. M. Kaldor, Relation between HIV viral load and infectiousness: A model-based analysis, Lancet, 372 (2008), 314-320. doi: 10.1016/S0140-6736(08)61115-0. Google Scholar

[44]

T. ZhangM. JiaH. LuoY. Zhou and N. Wang, Study on a HIV/AIDS model with application to Yunnan province, China, Appl Math Model, 35 (2011), 4379-4392. doi: 10.1016/j.apm.2011.03.004. Google Scholar

[45]

M. Zwahlen and M. Egger, Progression and Mortality of Untreated HIV-positive Individuals Living in Resource-Limited Settings: Update of Literature Review and Evidence Synthesis, Report to UNAIDS obligation no HQ/05/422204.Google Scholar

Figure 1.  Block diagram of the model
Figure 2.  Rate of immigration in Italy averaged over each year. Each time label denotes January 1st of the reported year
Figure 3.  Per capita loss rate averaged over each year
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
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)
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
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
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
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
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
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
Figure 12.  Infection-transfer graph $G$ of model 4. Transfers of individuals between compartments, black arcs; infections, red arcs
Table 1.  Baseline parameter values
ParametersValueSource
$\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)$$5,475$ daysAssumed
$1/\xi(t=2013)$$10,950$ daysAssumed
$\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}$$0.2$[34,6]
δ21.10·10-19 day-1Estimated
δ32.27·10-3 day-1Estimated
δ43.2·10-3 day-1Estimated
ParametersValueSource
$\bar{\Lambda}$$-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$ daysAssumed
$1/\xi(t=2013)$$10,950$ daysAssumed
$\theta_{1}$$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]
δ21.10·10-19 day-1Estimated
δ32.27·10-3 day-1Estimated
δ43.2·10-3 day-1Estimated
Table 2.  Predictions for different values of $\beta_5/\beta_2$
Values at January 1st 2025 $ {{\beta }_{5}}/{{\beta }_{2}} $
0.10.20.3
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$^{-1}$)3.855.7947.816
New cases of AIDS (persons$\cdot$year$^{-1}$)975.310621149
AIDS deaths (persons$\cdot$year$^{-1}$)557.3606.7656.8
Values at January 1st 2025 $ {{\beta }_{5}}/{{\beta }_{2}} $
0.10.20.3
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$^{-1}$)3.855.7947.816
New cases of AIDS (persons$\cdot$year$^{-1}$)975.310621149
AIDS deaths (persons$\cdot$year$^{-1}$)557.3606.7656.8
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