A Network Immuno-Epidemiological HIV Model

Abstract

In this paper we formulate a multi-scale nested immuno-epidemiological model of HIV on complex networks. The system is described by ordinary differential equations coupled with a partial differential equation. First, we prove the existence and uniqueness of the immunological model and then establish the well-posedness of the multi-scale model. We derive an explicit expression of the basic reproduction number \({\mathscr {R}}_{0}\) of the immuno-epidemiological model. The system has a disease-free equilibrium and an endemic equilibrium. The disease-free equilibrium is globally stable when \({\mathscr {R}}_{0}<1\) and unstable when \({\mathscr {R}}_0 >1\). Numerical simulations suggest that \({\mathscr {R}}_{0}\) increases as the number of nodes in the network increases. Further, we find that for a scale-free network the number of infected individuals at equilibrium is a hump-like function of the within-host reproduction number; however, the dependence becomes monotone if the network has predominantly low connectivity nodes or high connectivity nodes.

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Code availability

The code was written by one of the authors and is available on request.

References

  1. Acevedo MA, Prosper O, Lopiano K, Nick Ruktanonchai T, Caughlin T, Martcheva M, Osenberg CW, Smith DL (2015) Spatial heterogeneity, host movement and mosquito-borne disease transmission. PLoS ONE 10(6):e0127552

    Article  Google Scholar 

  2. Cuadros DF, Garcia-Ramos G (2012) Variable effect of co-infection on the HIV infectivity: within-host dynamics and epidemiological significance. Theor Biol Med Model. https://doi.org/10.1186/1742-4682-9-9

    Article  Google Scholar 

  3. De Leenheer P, Smith H (2003) Virus dynamics: a global analysis. SIAM J Appl Math 63(4):1313–1327

    MathSciNet  Article  Google Scholar 

  4. Doekes HM, Fraser C, Lythgoe KA (2017) Effect of the latent reservoir on the evolution of HIV at the within- and between-host levels. PLoS Comput Biol 13(1):1–27. https://doi.org/10.1371/journal.pcbi.1005228

    Article  Google Scholar 

  5. Gilchrist MA, Coombs D (2006) Evolution of virulence: interdependence, constraints, and selection using nested models. Theor Popul Biol 69:145–153

    Article  Google Scholar 

  6. Gilchrist M, Sasaki A (2002) Modeling host-parasite coevolution: a nested approach based on mechanistic models. J Theor Biol 218:289–308

    MathSciNet  Article  Google Scholar 

  7. Gulbudak H, Cannataro V, Tuncer N, Martcheva M (2017) Vector-borne pathogen and host evolution in a structured immuno-epidemiological system. Bull Math Biol 79:325–355

    MathSciNet  Article  Google Scholar 

  8. Gumel A, Mccluskey C, Driessche P (2006) Mathematical study of a staged-progression HIV model with imperfect vaccine. Bull Math Biol 68:2105–28. https://doi.org/10.1007/s11538-006-9095-7

    MathSciNet  Article  MATH  Google Scholar 

  9. https://www.unaids.org/en/resources/fact-sheet

  10. Huang G, Liu X, Takeuchi Y (2012) Lyapunov functions and global stability for age-structured HIV infection model. SIAM J Appl Math 72(1):25–38

    MathSciNet  Article  Google Scholar 

  11. Jin Z, Sun G, Zhu H (2014) Epidemic models for complex networks with demographics. Math Biosci Eng 11:1295–1317

    MathSciNet  Article  Google Scholar 

  12. Keeling MJ, Eames KTD (2005) Networks and epidemic models. J R Soc Interface 2:295–307

    Article  Google Scholar 

  13. Li C-H, Tsai C-C, Yang S-Y (2014) Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks. Commun Nonlinear Sci Numer Simul 19:1042–1054. https://doi.org/10.1016/j.cnsns.2013.08.033

    MathSciNet  Article  MATH  Google Scholar 

  14. Lythgoe KA, Pellis L, Fraser C (2013) Is HIV short-sighted? Insights from a multistrain nested model. Evolution 67(10):2769–2782. https://doi.org/10.1111/evo.12166

    Article  Google Scholar 

  15. Martcheva M, Li XZ (2013) Linking immunological and epidemiological dynamics of HIV: The case of super-infection. J Biol Dyn 7(1):161–182

    MathSciNet  Article  Google Scholar 

  16. Metzger VT, Lloyd-Smith JO, Weinberger LS (2011) Autonomous targeting of infectious superspreaders using engineered transmissible therapies. PLoS Comput Biol 7(3):1–12. https://doi.org/10.1371/journal.pcbi.1002015

    Article  Google Scholar 

  17. Nowak M, May R (2000) Virus dynamics: mathematical principles of immunology and virology. Oxford University Press, Oxford

    Google Scholar 

  18. Numfor E, Bhattacharya S, Lenhart S, Martcheva M (2014) Optimal control in coupled within-host and between-host models. Math Model Nat Phenom 9:171–203. https://doi.org/10.1051/mmnp/20149411

    MathSciNet  Article  MATH  Google Scholar 

  19. Numfor E, Bhattacharya S, Lenhart S, Martcheva M (2016) Optimal control in multi-group coupled within-host and between-host models. Electron J Differ Equ. Conf. 23:87–117

    MathSciNet  MATH  Google Scholar 

  20. Pastor-Satorras R, Vespignani A (2001a) Epidemic spreading in scale-free networks. Phys Rev Lett 86:3200–3. https://doi.org/10.1103/PhysRevLett.86.3200

    Article  Google Scholar 

  21. Pastor-Satorras R, Vespignani A (2001b) Epidemic dynamics and endemic states in complex networks. Phys Rev E Stat Nonlinear Soft Matter Phys 63:066117. https://doi.org/10.1103/PhysRevE.63.066117

    Article  Google Scholar 

  22. Perelson A, Nelson P (1999) Mathematical analysis of HIV-I: dynamics in vivo. SIAM Rev 41(1):3–44

    MathSciNet  Article  Google Scholar 

  23. Rong L, Feng Z, Perelson A (2007) Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy. SIAM J Appl Math 67:731–756. https://doi.org/10.1137/060663945

    MathSciNet  Article  MATH  Google Scholar 

  24. Rotenberg R (2009) HIV transmission networks. Curr Opin HIV/AIDS 4:260–265

    Article  Google Scholar 

  25. Ruth MA, Blower SM (1993) Imperfect vaccines and herd immunity to HIV 253. Proc R Soc Lond B. https://doi.org/10.1098/rspb.1993.0075

    Article  Google Scholar 

  26. Saenz RA, Bonhoeffer S (2013) Nested model reveals potential amplification of an HIV epidemic due to drug resistance. Epidemics 5(1):34–43

    Article  Google Scholar 

  27. Shen M, Xiao Y, Rong L (2015) Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics. Math Biosci 263:37–50. https://doi.org/10.1016/j.mbs.2015.02.003

    MathSciNet  Article  MATH  Google Scholar 

  28. Thieme HR (2003) Mathematics in population biology. Oxford University Press, Oxford

    Google Scholar 

  29. Thieme H, Castillo-Chavez C (1993) How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS? SIAM J Appl Math 53(5):1447–1479

    MathSciNet  Article  Google Scholar 

  30. Vieira IT, Cheng RCH, Harper PR et al (2010) Small world network models of the dynamics of HIV infection. Ann Oper Res 178:173–200. https://doi.org/10.1007/s10479-009-0571-y

    MathSciNet  Article  MATH  Google Scholar 

  31. Wang Y, Jin J, Yang Z, Zhang Z, Zhou T, Sun G (2012) Global analysis of an SIS model with an infective vector on complex networks. Nonlinear Anal Real World Appl 13:543–557

    MathSciNet  Article  Google Scholar 

  32. Wang Y, Cao J, Alofi A, Al Mazrooei A, Elaiw A (2015) Revisiting node-based SIR models in complex networks with degree correlations. Phys A Stat Mech Appl. https://doi.org/10.1016/j.physa.2015.05.103

    Article  MATH  Google Scholar 

  33. Yang J-Y, Chen Y (2017) Effect of infection age on an SIR epidemic model with demography on complex networks. Phys A Stat Mech Appl. https://doi.org/10.1016/j.physa.2017.03.006

    Article  Google Scholar 

  34. Yang J, Chen Y, Xu F (2016) Effect of infection age on an SIS epidemic model on complex networks. J Math Biol 73:1227–1249

    MathSciNet  Article  Google Scholar 

  35. Zhang J, Zhen J (2011) The analysis of an epidemic model on networks. Appl Math Comput 217:7053–7064

    MathSciNet  MATH  Google Scholar 

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Funding

Maia Martcheva was partially supported by NSF DMS-1951595. Necibe Tuncer was partially supported by NSF DMS-1951626.

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Correspondence to Churni Gupta or Necibe Tuncer or Maia Martcheva.

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Gupta, C., Tuncer, N. & Martcheva, M. A Network Immuno-Epidemiological HIV Model. Bull Math Biol 83, 18 (2021). https://doi.org/10.1007/s11538-020-00855-3

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Keywords

  • HIV
  • Network
  • Age structured
  • Basic reproduction number
  • Epidemic model