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On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition
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 |
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.
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. |
[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. |
[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. |
[13] |
H. W. Hethcote,
The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[32] |
D. Wanduku, Modeling highly random dynamical infectious systems, in Applied Mathematical Analysis: Theory, Methods, and Applications, Springer, (2020), 509–578. |
[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.
|
[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. |
[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 |
show all references
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. |
[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. |
[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. |
[13] |
H. W. Hethcote,
The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[32] |
D. Wanduku, Modeling highly random dynamical infectious systems, in Applied Mathematical Analysis: Theory, Methods, and Applications, Springer, (2020), 509–578. |
[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.
|
[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. |
[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 |




Parameter | Symbol(s) | Estimate(s) in years |
Effective response rate of |
0.1, 0.1, 0.8 | |
Infection transmission rates | 0.0211, 0.001055, 0.00844 | |
Natural death rates of |
0.01568 | |
Natural death rates of |
0.01568 | |
Infection related death rates of |
0.1474 | |
Infection related death rates of |
0.03685 | |
Infection related death rates of |
0.2948 | |
Recruitment rate | B | 0.55 |
Return rate from |
0.01 | |
Failure of treatment rate from |
0.01 | |
Proportion of newly infected individuals from the class |
0 - 1 | |
Time delay to progress to full blown AIDS | 2 - 15 | |
Time delay to begin treatment | 0.38-15 |
Parameter | Symbol(s) | Estimate(s) in years |
Effective response rate of |
0.1, 0.1, 0.8 | |
Infection transmission rates | 0.0211, 0.001055, 0.00844 | |
Natural death rates of |
0.01568 | |
Natural death rates of |
0.01568 | |
Infection related death rates of |
0.1474 | |
Infection related death rates of |
0.03685 | |
Infection related death rates of |
0.2948 | |
Recruitment rate | B | 0.55 |
Return rate from |
0.01 | |
Failure of treatment rate from |
0.01 | |
Proportion of newly infected individuals from the class |
0 - 1 | |
Time delay to progress to full blown AIDS | 2 - 15 | |
Time delay to begin treatment | 0.38-15 |
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