Modeling metapopulation dynamics of HIV epidemic on a linear lattice with nearest neighbour coupling

  • Authors

    • Titus Rotich Moi UniversityP.O. BOX 3900 - 30100ELDORET - KENYA
    • Robert Cheruiyot South Eastern Kenya University, P.O. Box 170, Kitui
    • Pauline Anupi Kisii University, P.O. Box 408-40200, KISII
    • Flomena Jeptanui University of Eldoret, P.O. Box 1125, Eldoret
    2016-03-02
    https://doi.org/10.14419/ijamr.v5i1.5544
  • Coupling Strength, Isolation, Metapopulation, Reproductive ratio, Synchronization.
  • Many mathematical models for the spread of infectious diseases in a population assume homogeneous mixing, but due to spatial distribution, there exist distinct patches with unique disease dispersion dynamics, especially if between patch mixing due to travel and migration is limited. In this paper, three levels of disease status in a - patch metapopulation was studied using a simple SIR-HIV epidemic model in a one dimensional nearest neighbour coupling lattice. The basic reproductive ratio , which is a function of coupling strength , is shown to affect stability characteristics of equilibrium points. The disease free equilibrium (DFE) is globally asymptotically stable irrespective of the value of  but the stability of the endemic equilibrium point (EEP) depends on the coupling strength . It was found that at the critical value of coupling strength , the subpopulations dynamics are synchronized while for  the subpopulation dynamics are independent. Patch isolation strategy for the control of HIV dispersion requires a critical coupling strength of . This interaction restriction reduces  to values less than one, and the disease will be eliminated, making isolation effective. Demographic and epidemiological parameters of Vihiga County in Kenya were used in the study.

    Author Biography

    • Titus Rotich, Moi UniversityP.O. BOX 3900 - 30100ELDORET - KENYA

      Head of Subject

      Mathematics Department

  • References

    1. [1] J. Arino and P. Van Den Driessche, Disease spread in metapopulations, Nonlinear dynamics and evolution equations, 48 (2006), pp. 1-13.

      [2] F. Baryarama, L. S. Luboobi and J. Y. Mugisha, Periodicity of the HIV/AIDS epidemic in a mathematical model that incorporates complacency, Am. J. Infect. Dis, 1 (2005), pp. 55-60. http://dx.doi.org/10.3844/ajidsp.2005.55.60.

      [3] M. Bawa, S. Abdulrahman, O. Jimoh and N. Adabara, STABILITY ANALYSIS OF THE DISEASE-FREE EQUILIBRIUM STATE FOR LASSA FEVER DISEASE, Journal of Science, Technology, Mathematics and Education (JOSTMED), 9 (2013), pp. 115-123.

      [4] D. Brockmann, L. Hufnagel and T. Geisel, The scaling laws of human travel, Nature, 439 (2006), pp. 462-465. http://dx.doi.org/10.1038/nature04292.

      [5] D. H. Brown and B. M. Bolker, The effects of disease dispersal and host clustering on the epidemic threshold in plants, Bulletin of mathematical biology, 66 (2004), pp. 341-371. http://dx.doi.org/10.1016/j.bulm.2003.08.006.

      [6] V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, Journal of theoretical biology, 251 (2008), pp. 450-467. http://dx.doi.org/10.1016/j.jtbi.2007.11.028.

      [7] R. V. Culshaw, Mathematical modeling of AIDS progression: limitations, expectations, and future directions, Journal of American Physicians and Surgeons, 11 (2006), pp. 101.

      [8] H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Progress of Theoretical Physics, 69 (1983), pp. 32-47. http://dx.doi.org/10.1143/PTP.69.32.

      [9] L. Hufnagel, D. Brockmann and T. Geisel, Forecast and control of epidemics in a globalized world, Proceedings of the National Academy of Sciences of the United States of America, 101 (2004), pp. 15124-15129. http://dx.doi.org/10.1073/pnas.0308344101.

      [10] M. Jesse, P. Ezanno, S. Davis and J. Heesterbeek, A fully coupled, mechanistic model for infectious disease dynamics in a metapopulation: movement and epidemic duration, Journal of theoretical biology, 254 (2008), pp. 331-338. http://dx.doi.org/10.1016/j.jtbi.2008.05.038.

      [11] M. Jesse and H. Heesterbeek, Divide and conquer? Persistence of infectious agents in spatial metapopulations of hosts, Journal of theoretical biology, 275 (2011), pp. 12-20. http://dx.doi.org/10.1016/j.jtbi.2011.01.032.

      [12] R. Levins, T. Awerbuch, U. Brinkmann, I. Eckardt, P. Epstein, N. Makhoul, C. A. De Possas, C. Puccia, A. Spielman and M. E. Wilson, The emergence of new diseases, American Scientist, 82 (1994), pp. 52-60.

      [13] A. L. Lloyd and V. A. Jansen, Spatiotemporal dynamics of epidemics: synchrony in metapopulation models, Mathematical biosciences, 188 (2004), pp. 1-16. http://dx.doi.org/10.1016/j.mbs.2003.09.003.

      [14] D. Okuonghae and R. Okuonghae, A mathematical model for Lassa fever, Journal of the Nigerian Association of Mathematical Physics, 10 (2008).

      [15] R. Patel, I. M. Longini and M. E. Halloran, Finding optimal vaccination strategies for pandemic influenza using genetic algorithms, Journal of theoretical biology, 234 (2005), pp. 201-212. http://dx.doi.org/10.1016/j.jtbi.2004.11.032.

      [16] C. M. Rivers, E. T. Lofgren, M. Marathe, S. Eubank and B. L. Lewis, Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia, PLoS currents, 6 (2014). http://dx.doi.org/10.1371/currents.outbreaks.4d41fe5d6c05e9df30ddce33c66d084c.

      [17] M. Roberts and J. Heesterbeek, Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection, Journal of mathematical biology, 55 (2007), pp. 803-816. http://dx.doi.org/10.1007/s00285-007-0112-8.

      [18] A. Thowsen, The Routh-Hurwitz method for stability determination of linear differential-difference systems†, International Journal of Control, 33 (1981), pp. 991-995. http://dx.doi.org/10.1080/00207178108922971.

      [19] S. Towers, O. Patterson-Lomba and C. Castillo-Chavez, Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak, PLOS Currents Outbreaks, 6 (2014).

      [20] A. A. Wasike, Synchronization and oscillator death in diffusively coupled lattice oscillators, African Journal of Science and Technology, 4 (2003).

      [21] A. A. Wasike and K. Rotich, Synchronization and persistence in Diffusively Coupled Lattice Oscillators, (2007).

  • Downloads

  • How to Cite

    Rotich, T., Cheruiyot, R., Anupi, P., & Jeptanui, F. (2016). Modeling metapopulation dynamics of HIV epidemic on a linear lattice with nearest neighbour coupling. International Journal of Applied Mathematical Research, 5(1), 73-83. https://doi.org/10.14419/ijamr.v5i1.5544