# American Institute of Mathematical Sciences

## Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays

 Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave, Room 3309, Statesboro, Georgia, 30460, USA

* Corresponding author

Received  July 2020 Revised  October 2020 Published  January 2021

A multipopulation HIV/AIDS deterministic epidemic model is studied. The population structure is a multihuman behavioral structure composed of humans practicing varieties of distinct HIV/AIDS preventive measures learnt from information and education campaigns (IEC) in the community. Antiretroviral therapy (ART) treatment is considered, and the delay from HIV exposure until the onset of ART is considered. The effects of national and multilateral support providing official developmental assistance (ODAs) to combat HIV are represented. A separate dynamics for the IEC information density in the community is derived. The epidemic model is a system of differential equations with random delays. The basic reproduction number (BRN) for the dynamics is obtained, and stability analysis of the system is conducted, whereby other disease control conditions are obtained in a multi- and a finite dimensional phase space. Numerical simulation results are given.

Citation: Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021005
##### References:
 [1] World Data Atlas: Uganda - Crude Death Rate, 2020. Available from: https://www.knoema.com/atlas/uganda/death-rate. Google Scholar [2] CDC: About HIV, 2020. Available from: https://www.cdc.gov/hiv/basics/whatishiv.html. Google Scholar [3] The Global Fund, 2020. Available from: https://www.theglobalfund.org/en/. Google Scholar [4] Healio, HIV/AIDS, Infectious Disease News: Cuts in Foreign Aid for HIV Place Millions at Risk, 2017. Available from: https://www.healio.com/news/infectious-disease/20171010/cuts-in-foreign-aid-for-hiv-place-millions-at-risk. Google Scholar [5] HIV.gov, Symptoms of HIV: How Can You Tell if You Have Hiv?, 2020. Available from: https://www.hiv.gov/hiv-basics/overview/about-hiv-and-aids/symptoms-of-hiv. Google Scholar [6] WHO, HIV/AIDS: Pre-Exposure Prophylaxis, 2020. Available from: https://www.who.int/hiv/topics/prep/en/. Google Scholar [7] CDC: PrEP (Pre-Exposure Prophylaxis), 2020. Available from: https://www.cdc.gov/hiv/basics/prep.html. Google Scholar [8] HIV.gov: Presidendent's Emergency Plan for Aids Relief, 2020. Available from: https://www.hiv.gov/federal-response/pepfar-global-aids/pepfar. Google Scholar [9] WHO: HIV/AIDS, 2020. Available from: https://www.who.int/news-room/fact-sheets/detail/hiv-aids. Google Scholar [10] S. Del Valle, A. Morales Evangelista, M. C. Velasco, C. Kribs-Zaleta and S.-F. Hsu Schmitz, Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity, Math. Biosci., 187 (2004), 111-133.  doi: 10.1016/j.mbs.2003.11.004.  Google Scholar [11] J. Fobil and I. Soyiri, An assessment of government policy response to HIV/AIDS in ghana, Sahara J-J Soc Asp H, 3 (2006), 457-465.  doi: 10.1080/17290376.2006.9724872.  Google Scholar [12] E. C. Green, D. T. Halperin, V. Nantulya and J. A. Hogle, Uganda's HIV prevention success: the role of sexual behavior change and the national response, AIDS Behav., 10 (2006), 335-346.  doi: 10.1007/s10461-006-9073-y.  Google Scholar [13] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar [14] H.-F. Huo, R. Chen and X.-Y. Wang, Modelling and stability of HIV/AIDS epidemic model with treatment, Appl. Math. Model., 40 (2016), 6550-6559.  doi: 10.1016/j.apm.2016.01.054.  Google Scholar [15] H. Joshi, S. Lenhart, K. Albright and K. Gipson, Modeling the effect of information campaigns on the HIV epidemic in Uganda, Math. Biosci. Eng., 5 (2008), 757-770.  doi: 10.3934/mbe.2008.5.757.  Google Scholar [16] I. Kasamba, K. Baisley, B. N. Mayanja, D. Maher and H. Grosskurth, The impact of antiretroviral treatment on mortality trends of HIV-positive adults in rural Uganda: A longitudinal population-based study, 1999–2009, Trop. Med. Int. Health, 17 (2012), e66–e73. doi: 10.1111/j.1365-3156.2012.02841.x.  Google Scholar [17] A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128.  doi: 10.1093/imammb/dqi001.  Google Scholar [18] A. Kumar, P. K. Srivastava and Y. Takeuchi, Modeling the role of information and limited optimal treatment on disease prevalence, J. Theoret. Biol., 414 (2017), 103-119.  doi: 10.1016/j.jtbi.2016.11.016.  Google Scholar [19] S. D. Lawn, M. E. Török and R. Wood, Optimum time to start antiretroviral therapy during hiv-associated opportunistic infections, Curr. Opin. Infect. Dis., 24 (2011), 34-42.  doi: 10.1097/QCO.0b013e3283420f76.  Google Scholar [20] H. Liu and J.-F. Zhang, Dynamics of two time delays differential equation model to hiv latent infection, Phys. A, 514 (2019), 384-395.  doi: 10.1016/j.physa.2018.09.087.  Google Scholar [21] S.-H. Ma and H.-F. Huo, Global dynamics for a multi-group alcoholism model with public health education and alcoholism age, Math. Biosci. Eng., 16 (2019), 1683-1708.  doi: 10.3934/mbe.2019080.  Google Scholar [22] Z. Mukandavire, W. Garira and J. M. Tchuenche, Modelling effects of public health educational campaigns on HIV/AIDS transmission dynamics, Appl. Math. Model., 33 (2009), 2084-2095.  doi: 10.1016/j.apm.2008.05.017.  Google Scholar [23] P. Nunnenkamp and H. Öhler, Throwing foreign aid at HIV/AIDS in developing countries: Missing the target, World Dev., 39 (2011), 1704-1723.   Google Scholar [24] S. Okware, J. Kinsman, S. Onyango and et. al., Revisiting the ABC strategy: HIV prevention in Uganda in the era of antiretroviral therapy, Postgrad Med. J., 81 (2005), 625–628. doi: 10.1136/pgmj.2005.032425.  Google Scholar [25] S. Singh, J. E. Darroch and A. Bankole, A, b and c in Uganda: The roles of abstinence, monogamy and condom use in HIV decline, Reprod. Health Matters, 12 (2004), 129-131.  doi: 10.1016/S0968-8080(04)23118-4.  Google Scholar [26] UNAIDS, Making Condoms Work for HIV Prevention. Cutting-edge Perspectives. UNAIDS Best Practice Collection, 2004. Google Scholar [27] R. P. Walensky, E. D. Borre, L.-G. Bekker, E. P. Hyle, G. S. Gonsalves, R. Wood, S. P. Eholié, M. C. Weinstein, X. Anglaret, K. A. Freedberg and et. al., Do less harm: Evaluating HIV programmatic alternatives in response to cutbacks in foreign aid, Ann. Intern. Med., 167 (2017), 618–629. doi: 10.7326/M17-1358.  Google Scholar [28] D. Wanduku, The stationary distribution and stochastic persistence for a class of disease models: Case study of malaria, Int. J. of Biomath., 13 (2020), 2050024, 59 pp. doi: 10.1142/S1793524520500242.  Google Scholar [29] D. Wanduku, Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbations, Appl. Math. Comput., 294 (2017), 49-76.  doi: 10.1016/j.amc.2016.09.001.  Google Scholar [30] D. Wanduku, Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment, Int. J. of Biomath., 11 (2018), 1850085, 46 pp. doi: 10.1142/S1793524518500857.  Google Scholar [31] D. Wanduku, The stochastic extinction and stability conditions for nonlinear malaria epidemics, Math. Biosci. Eng., 16 (2019), 3771-3806.  doi: 10.3934/mbe.2019187.  Google Scholar [32] D. Wanduku, Modeling highly random dynamical infectious systems, in Applied Mathematical Analysis: Theory, Methods, and Applications, Springer, (2020), 509–578.  Google Scholar [33] D. Wanduku, A nonlinear multi-population behavioral model to assess the roles of education campaigns, random supply of AIDS, and delayed art treament in HIV/AIDS epidemics, Math. Biosci. Eng., 17 (2020), 6791-6837.   Google Scholar [34] D. Wanduku, On the almost sure convergence of a stochastic process in in a class of nonlinear multi-population behavioral models for hiv/aids with delayed art treatment, to appear in, Stoch Anal Appl.. Google Scholar [35] D. Wanduku and G. S. Ladde, Fundamental properties of a two-scale network stochastic human epidemic dynamic model, Neural Parallel Sci. Comput., 19 (2011), 229-269.   Google Scholar [36] D. Wanduku and B. O. Oluyede, Some asymptotic properties of SEIRS models with nonlinear incidence and random delays, Nonlinear Anal. Model. Control, 25 (2020), 461-481.  doi: 10.15388/namc.2020.25.16660.  Google Scholar [37] WHO, WHO expands recommendation on oral pre-exposure prophylaxis of HIV infection (PrEP), policy brief, WHO Reference Number: WHO/HIV/2015.48, 1–2. Google Scholar

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##### References:
 [1] World Data Atlas: Uganda - Crude Death Rate, 2020. Available from: https://www.knoema.com/atlas/uganda/death-rate. Google Scholar [2] CDC: About HIV, 2020. Available from: https://www.cdc.gov/hiv/basics/whatishiv.html. Google Scholar [3] The Global Fund, 2020. Available from: https://www.theglobalfund.org/en/. Google Scholar [4] Healio, HIV/AIDS, Infectious Disease News: Cuts in Foreign Aid for HIV Place Millions at Risk, 2017. Available from: https://www.healio.com/news/infectious-disease/20171010/cuts-in-foreign-aid-for-hiv-place-millions-at-risk. Google Scholar [5] HIV.gov, Symptoms of HIV: How Can You Tell if You Have Hiv?, 2020. Available from: https://www.hiv.gov/hiv-basics/overview/about-hiv-and-aids/symptoms-of-hiv. Google Scholar [6] WHO, HIV/AIDS: Pre-Exposure Prophylaxis, 2020. Available from: https://www.who.int/hiv/topics/prep/en/. Google Scholar [7] CDC: PrEP (Pre-Exposure Prophylaxis), 2020. Available from: https://www.cdc.gov/hiv/basics/prep.html. Google Scholar [8] HIV.gov: Presidendent's Emergency Plan for Aids Relief, 2020. Available from: https://www.hiv.gov/federal-response/pepfar-global-aids/pepfar. Google Scholar [9] WHO: HIV/AIDS, 2020. Available from: https://www.who.int/news-room/fact-sheets/detail/hiv-aids. Google Scholar [10] S. Del Valle, A. Morales Evangelista, M. C. Velasco, C. Kribs-Zaleta and S.-F. Hsu Schmitz, Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity, Math. Biosci., 187 (2004), 111-133.  doi: 10.1016/j.mbs.2003.11.004.  Google Scholar [11] J. Fobil and I. Soyiri, An assessment of government policy response to HIV/AIDS in ghana, Sahara J-J Soc Asp H, 3 (2006), 457-465.  doi: 10.1080/17290376.2006.9724872.  Google Scholar [12] E. C. Green, D. T. Halperin, V. Nantulya and J. A. Hogle, Uganda's HIV prevention success: the role of sexual behavior change and the national response, AIDS Behav., 10 (2006), 335-346.  doi: 10.1007/s10461-006-9073-y.  Google Scholar [13] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar [14] H.-F. Huo, R. Chen and X.-Y. Wang, Modelling and stability of HIV/AIDS epidemic model with treatment, Appl. Math. Model., 40 (2016), 6550-6559.  doi: 10.1016/j.apm.2016.01.054.  Google Scholar [15] H. Joshi, S. Lenhart, K. Albright and K. Gipson, Modeling the effect of information campaigns on the HIV epidemic in Uganda, Math. Biosci. Eng., 5 (2008), 757-770.  doi: 10.3934/mbe.2008.5.757.  Google Scholar [16] I. Kasamba, K. Baisley, B. N. Mayanja, D. Maher and H. Grosskurth, The impact of antiretroviral treatment on mortality trends of HIV-positive adults in rural Uganda: A longitudinal population-based study, 1999–2009, Trop. Med. Int. Health, 17 (2012), e66–e73. doi: 10.1111/j.1365-3156.2012.02841.x.  Google Scholar [17] A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128.  doi: 10.1093/imammb/dqi001.  Google Scholar [18] A. Kumar, P. K. Srivastava and Y. Takeuchi, Modeling the role of information and limited optimal treatment on disease prevalence, J. Theoret. Biol., 414 (2017), 103-119.  doi: 10.1016/j.jtbi.2016.11.016.  Google Scholar [19] S. D. Lawn, M. E. Török and R. Wood, Optimum time to start antiretroviral therapy during hiv-associated opportunistic infections, Curr. Opin. Infect. Dis., 24 (2011), 34-42.  doi: 10.1097/QCO.0b013e3283420f76.  Google Scholar [20] H. Liu and J.-F. Zhang, Dynamics of two time delays differential equation model to hiv latent infection, Phys. A, 514 (2019), 384-395.  doi: 10.1016/j.physa.2018.09.087.  Google Scholar [21] S.-H. Ma and H.-F. Huo, Global dynamics for a multi-group alcoholism model with public health education and alcoholism age, Math. Biosci. Eng., 16 (2019), 1683-1708.  doi: 10.3934/mbe.2019080.  Google Scholar [22] Z. Mukandavire, W. Garira and J. M. Tchuenche, Modelling effects of public health educational campaigns on HIV/AIDS transmission dynamics, Appl. Math. Model., 33 (2009), 2084-2095.  doi: 10.1016/j.apm.2008.05.017.  Google Scholar [23] P. Nunnenkamp and H. Öhler, Throwing foreign aid at HIV/AIDS in developing countries: Missing the target, World Dev., 39 (2011), 1704-1723.   Google Scholar [24] S. Okware, J. Kinsman, S. Onyango and et. al., Revisiting the ABC strategy: HIV prevention in Uganda in the era of antiretroviral therapy, Postgrad Med. J., 81 (2005), 625–628. doi: 10.1136/pgmj.2005.032425.  Google Scholar [25] S. Singh, J. E. Darroch and A. Bankole, A, b and c in Uganda: The roles of abstinence, monogamy and condom use in HIV decline, Reprod. Health Matters, 12 (2004), 129-131.  doi: 10.1016/S0968-8080(04)23118-4.  Google Scholar [26] UNAIDS, Making Condoms Work for HIV Prevention. Cutting-edge Perspectives. UNAIDS Best Practice Collection, 2004. Google Scholar [27] R. P. Walensky, E. D. Borre, L.-G. Bekker, E. P. Hyle, G. S. Gonsalves, R. Wood, S. P. Eholié, M. C. Weinstein, X. Anglaret, K. A. Freedberg and et. al., Do less harm: Evaluating HIV programmatic alternatives in response to cutbacks in foreign aid, Ann. Intern. Med., 167 (2017), 618–629. doi: 10.7326/M17-1358.  Google Scholar [28] D. Wanduku, The stationary distribution and stochastic persistence for a class of disease models: Case study of malaria, Int. J. of Biomath., 13 (2020), 2050024, 59 pp. doi: 10.1142/S1793524520500242.  Google Scholar [29] D. Wanduku, Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbations, Appl. Math. Comput., 294 (2017), 49-76.  doi: 10.1016/j.amc.2016.09.001.  Google Scholar [30] D. Wanduku, Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment, Int. J. of Biomath., 11 (2018), 1850085, 46 pp. doi: 10.1142/S1793524518500857.  Google Scholar [31] D. Wanduku, The stochastic extinction and stability conditions for nonlinear malaria epidemics, Math. Biosci. Eng., 16 (2019), 3771-3806.  doi: 10.3934/mbe.2019187.  Google Scholar [32] D. Wanduku, Modeling highly random dynamical infectious systems, in Applied Mathematical Analysis: Theory, Methods, and Applications, Springer, (2020), 509–578.  Google Scholar [33] D. Wanduku, A nonlinear multi-population behavioral model to assess the roles of education campaigns, random supply of AIDS, and delayed art treament in HIV/AIDS epidemics, Math. Biosci. Eng., 17 (2020), 6791-6837.   Google Scholar [34] D. Wanduku, On the almost sure convergence of a stochastic process in in a class of nonlinear multi-population behavioral models for hiv/aids with delayed art treatment, to appear in, Stoch Anal Appl.. Google Scholar [35] D. Wanduku and G. S. Ladde, Fundamental properties of a two-scale network stochastic human epidemic dynamic model, Neural Parallel Sci. Comput., 19 (2011), 229-269.   Google Scholar [36] D. Wanduku and B. O. Oluyede, Some asymptotic properties of SEIRS models with nonlinear incidence and random delays, Nonlinear Anal. Model. Control, 25 (2020), 461-481.  doi: 10.15388/namc.2020.25.16660.  Google Scholar [37] WHO, WHO expands recommendation on oral pre-exposure prophylaxis of HIV infection (PrEP), policy brief, WHO Reference Number: WHO/HIV/2015.48, 1–2. Google Scholar
Shows the behaviors of the modified standard incidence and the ordinary standard incidence rates as the number of infectives continually increase over time. Clearly, the modified standard incidence is more suitable for many real life scenarios where the incidence rate of the disease saturates over time as the number of infections increase in the population
shows the different states of the population in the HIV/AIDS epidemic, and the transition rates between the states. Note that the operators $\mathbb{E}_{\tau_{1}}[\cdots]$ and $\mathbb{E}_{\tau_{2}}[\cdots]$ represent expectations with respect to the random variables $\tau_{1}$ and $\tau_{2}$, respectively
(ⅰ) depicts a continuously rising relationship between the BRN and the delay $\tau_{2}$. (ⅱ) shows a declining relationship between the BRN and the proportion $1-\varepsilon_{0}$ of individuals who are receiving ART treatment
(a-1)-(d-1) shows the behavior of the path of the total susceptible population $S(t) = S_{0}(t)+S_{1}(t)+S_{2}(t)$, over time, whenever the conditions of Theorem 5.3 and Theorem 5.4 are satisfied. Clearly, the path of $S(t)$ is persistent as proven in Theorem 5.4 and approaches the DFE state $S^{*}_{0} = \frac{B}{\mu_{S_{0}}} = 26.13417$. The dotted redline in (d-1) is the value of $S^{*}_{0} = 26.13417$. The figures (e-1), (f-1) and (g-1) also show the paths of the HIV related states $I, T$ and $A$. Clearly, the paths of $I, T$ and $A$ approach the corresponding coordinate $0$ of the DFE $E_{0}$. In other words, the disease is getting extinct over time
Shows the list of model parameters, estimates and their definitions. Note that the parameters are expressed in years and converted to days for all simulations in Section 7
 Parameter Symbol(s) Estimate(s) in years Effective response rate of $S_0(t), S_1(t), S_2(t)$ $\gamma_0, \gamma_1, \gamma_2$ 0.1, 0.1, 0.8 Infection transmission rates $\beta_0, \beta_1, \beta_2$ 0.0211, 0.001055, 0.00844 Natural death rates of $S_0(t), S_1(t), S_2(t)$ $\mu_{S_0}, \mu_{S_1}, \mu_{S_2}$ 0.01568 Natural death rates of $I(t), T(t), A(t), R(t)$ $\mu_I, \mu_T, \mu_A, \mu_R$ 0.01568 Infection related death rates of $I(t)$ $d_I$ 0.1474 Infection related death rates of $T(t)$ $d_T$ 0.03685 Infection related death rates of $A(t)$ $d_A$ 0.2948 Recruitment rate B 0.55 Return rate from $T(t)$ to $I(t)$ $\alpha_{TI}$ 0.01 Failure of treatment rate from $T(t)$ to $A(t)$ $\alpha_{TA}$ 0.01 Proportion of newly infected individuals from the class $S_j, j=0, 1, 2$ who do not receive ART and joins full blown AIDS state $A(t)$ $\epsilon_0, \epsilon_1, \epsilon_2$ 0 - 1 Time delay to progress to full blown AIDS $\tau_1$ 2 - 15 Time delay to begin treatment $\tau_2$ 0.38-15
 Parameter Symbol(s) Estimate(s) in years Effective response rate of $S_0(t), S_1(t), S_2(t)$ $\gamma_0, \gamma_1, \gamma_2$ 0.1, 0.1, 0.8 Infection transmission rates $\beta_0, \beta_1, \beta_2$ 0.0211, 0.001055, 0.00844 Natural death rates of $S_0(t), S_1(t), S_2(t)$ $\mu_{S_0}, \mu_{S_1}, \mu_{S_2}$ 0.01568 Natural death rates of $I(t), T(t), A(t), R(t)$ $\mu_I, \mu_T, \mu_A, \mu_R$ 0.01568 Infection related death rates of $I(t)$ $d_I$ 0.1474 Infection related death rates of $T(t)$ $d_T$ 0.03685 Infection related death rates of $A(t)$ $d_A$ 0.2948 Recruitment rate B 0.55 Return rate from $T(t)$ to $I(t)$ $\alpha_{TI}$ 0.01 Failure of treatment rate from $T(t)$ to $A(t)$ $\alpha_{TA}$ 0.01 Proportion of newly infected individuals from the class $S_j, j=0, 1, 2$ who do not receive ART and joins full blown AIDS state $A(t)$ $\epsilon_0, \epsilon_1, \epsilon_2$ 0 - 1 Time delay to progress to full blown AIDS $\tau_1$ 2 - 15 Time delay to begin treatment $\tau_2$ 0.38-15
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