TWO-PATCH SIS STOCHASTIC MODEL: EFFECTS OF DISPERSAL RATES ON DISEASE TRANSMISSION

Authors

  • Auwal Abdullahi
  • Babale Aliyu

Abstract

Communicable diseases including measles, Covid-19 and Ebola virus can be transmitted from one community to the other through epidemic dispersal. Different dispersal mechanisms were investigated using a verity of mathematical models; however, the effect of dispersal rates on diseases transmitting between communities that differ in healthcare provisions has not been previously studied. This study, therefore, investigated such effects on the transmission of infectious diseases between two distinct patches: a community with (without) better healthcare facilities. The stochastic susceptible-infected-susceptible (SIS) model, devised through the continuous-time Markov chain (CTMC) process, together with its corresponding ordinary differential equation (ODE) model, was used to determine how changes in dispersal rates can affect the transmissions of diseases. To supplement the findings of this study, basic reproduction numbers for the two patches were also determined. We found that the dispersal rate has profound effects on the transmission of infectious diseases since increase in the dispersal rate in one community an increases the disease transmission in the other and the opposite is also true. Therefore, the transmission of diseases in not severely affected communities can be contained when travel ban is imposed in the worst affected communities.

Keywords: Continuous-time Markov Chain; Epidemic Dispersal; Gillespie Algorithm; Basic Reproduction Number

Author Biographies

Auwal Abdullahi

*1Department of Mathematics and Computer Science, Federal University Kashere, Nigeria

Babale Aliyu

2Department of Botany, Faculty of Science, Gombe State University, Nigeria

 

DOI: https://doi.org/10.5281/zenodo.8414385 

References

Hogben H, & Leichliter. JS. Social determinants and sexually transmitted disease disparities. Sexually transmitted diseases, pages 2008: S13–S18.

Yudkin JS, Holt RIG, Silva-Matos, C, & Beran D. Twinning for better diabetes care: a model for improving healthcare for non-communicable diseases in resource-poor countries, 2009.

Fallah, MP, Skrip, LA, Gertler S, Dan Yamin D, & Galvani AP. Quantifying poverty as a driver of ebola transmission. PLoS Neglected Tropical Diseases, 9(12):e0004260, 2015.

Kraemer MUG, Hay SI, Pigott DM, Smith DL, Wint GRW, & Golding N. Progress and challenges in infectious disease cartography. Trends in Parasitology, 2016:32(1), 19–29.

Bichara D, Kang Y, Castillo-Chavez C, Horan R, & Perrings C. Sis and sir epidemic models under virtual dispersal. Bulletin of Mathematical Biology, 2015:77, 2004–2034.

Wendi Wang W, & Zhao X-Q. An epidemic model with population dispersal and infection period. SIAM Journal on Applied Mathematics, 2006: 66(4), 1454–1472.

Wendi Wang & Xiao-Qiang Zhao. An epidemic model in a patchy environment. Mathematical biosciences, 2004: 190(1), 97–112.

Milliken E. The probability of extinction of infectious salmon anemia virus in one and two patches. Bulletin of Mathematical Biology, 2017:79(12), 2887–2904.

Fabre F, Coville J, & Cunniffe NJ. Optimising reactive disease management using spatially explicit models at the landscape scale. In Plant Diseases and Food Security in the 21st Century, pages 47–72. Springer, 2021.

Yang F-Y, Li Y, Li W-T, & Wang Z-C. Traveling waves in a nonlocal dispersal kermack-mckendrick epidemic model. Discrete & Continuous Dynamical Systems-Series B, 2013:18(7).

Mundt CC, Sackett KE, Wallace LD, Cowger C, & Dudley JP. Aerial dispersal and multiple-scale spread of epidemic disease. EcoHealth,2009: 6:546–552.

Filipe JAN, & Maule MM. Effects of dispersal mechanisms on spatio-temporal development of epidemics. Journal of Theoretical Biology, 2004:226(2):125–141.

Brown DH, & Bolker BM. The effects of disease dispersal and host clustering on the epidemic threshold in plants. Bulletin of Mathematical Biology, 2004: 66:341–371.

Xu X-M, & Ridout MS. Effects of initial epidemic conditions, sporulation rate, and spore dispersal gradient on the spatio-temporal dynamics of plant disease epidemics. Phytopathology, 1998:88(10):1000–1012.

Sackett KE, & Mundt CC. The effects of dispersal gradient and pathogen life cycle components on epidemic velocity in computer simulations. Phytopathology, 2005:95(9):992–1000.

Keeling MJ, Woolhouse MEJ, Shaw DJ, Matthews L, Chase-Topping M, Haydon DT, Cornell SJ, Kappey J, Wilesmith J, & Grenfell TB. Dynamics of the 2001 Uk foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape. Science, 2001:294(5543):813–817.

Hurtado PJ. Building new models: Rethinking and revising ode model assumptions. An Introduction to Undergraduate Research in Computational and Mathematical Biology: From Birdsongs to Viscosities, pages 1–86, 2020.

Allen LJS. A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis. Infectious Disease Modelling, 2017: 2(2),128–142.

Driessche PV, & Yakubu A. Disease extinction versus persistence in discrete-time epidemic models. Bulletin of Mathematical Biology, 81(11), 4412– 4446.

Liang X, Lei Zhang L, & Zhao X-Q. Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for lyme disease). Journal of Dynamics and Differential Equations, 2019: 31, 1247–1278.

Kot M. Elements of mathematical ecology. Cambridge University Press, 2001.

Grimmett G, & Stirzaker D. Probability and random processes. Oxford University Press, 2020.

Meakin SR, & Keeling MJ. Correlations between stochastic epidemics in two interacting populations. Epidemics, 2019:26, 58–67.

Nandi A, & Allen LJS. Stochastic two-group models with transmission dependent on host infectivity or susceptibility. Journal of Biological Dynamics, 2019:13(sup1), 201–224.

Liu Q, Jiang D, Hayat T, & Alsaedi A. Dynamics of a stochastic multigroup siqr epidemic model with standard incidence rates. Journal of the Franklin Institute, 2019:356(5):2960–2993.

Montagnon P. A stochastic sir model on a graph with epidemiological and population dynamics occurring over the same time scale. Journal of Mathematical Biology, 2019:79:31–62.

McKane AJ, & Newman TJ. Predator-prey cycles from resonant amplification of demographic stochasticity. Physical Review Letters, 2005:94(21), 218102.

Black AJ & McKane AJ. Stochastic formulation of ecological models and their applications. Trends in Ecology & Evolution, 2012:27(6):337–345.

Erban R, Chapman J, & Maini P. A practical guide to stochastic simulations of reaction-diffusion processes. arXiv preprint arXiv: 2007:0704.1908.

Bailey NTJ. The elements of stochastic processes with applications to the natural sciences, volume 25. John Wiley & Sons, 1991.

Brauer F, Van den Driessche P, Wu J, & Allen LJS. Mathematical epidemiology, volume 1945. Springer, 2008.

Delamater PL, Street EJ, Leslie TF, Yang YT, & Jacobsen KH. Complexity of the basic reproduction number (r0). Emerging Infectious Diseases, 2019:25(1), 1.

Brightwell G, House T, & Luczak M. Extinction times in the subcritical stochastic sis logistic epidemic. Journal of Mathematical Biology, 2018:77, 455–493.

Lahodny GE, & Allen LJS. Probability of a disease outbreak in stochastic multipatch epidemic models. Bulletin of Mathematical Biology, 2013:75, 1157–1180.

Gillespie DT. Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry,1977:81(25), 2340–2361.

Allen LJS, Wesley CL, Owen RD, Goodin DG, Koch D, Jonsson CB, Chu Y-K, Hutchinson JMS, & Paige RL. A habitat-based model for the spread of hantavirus between reservoir and spillover species. Journal of Theoretical Biology, 2009: 260(4), 510–522.

Lewis MA, Renclawowicz J, van den Driessche P, & Wonham M. A comparison of continuous and discrete-time west Nile virus models. Bulletin of Mathematical Biology, 2006:68, 491–509.

Linda JS Allen. An introduction to stochastic processes with applications to biology. CRC press, 2010.

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Published

2023-10-06