Impact of human mobility on the transmission dynamics of infectious diseases

Abstract

Spatial heterogeneity is an important aspect to be studied in infectious disease models. It takes two forms: one is local, namely diffusion in space, and other is related to travel. With the advancement of transportation system, it is possible for diseases to move from one place to an entirely separate place very quickly. In a developing country like India, the mass movement of large numbers of individuals creates the possibility of spread of common infectious diseases. This has led to the study of infectious disease model to describe the infection during transport. An SIRS-type epidemic model is formulated to illustrate the dynamics of such infectious disease propagation between two cities due to population dispersal. The most important threshold parameter, namely the basic reproduction number, is derived, and the possibility of existence of backward bifurcation is examined, as the existence of backward bifurcation is very unsettling for disease control and it is vital to know from modeling analysis when it can occur. It is shown that dispersal of populations would make the disease control difficult in comparison with nondispersal case. Optimal vaccination and treatment controls are determined. Further to find the best cost-effective strategy, cost-effectiveness analysis is also performed. Though it is not a case study, simulation work suggests that the proposed model can also be used in studying the SARS epidemic in Hong Kong, 2003.

Keywords

SIRS epidemic model / Basic reproduction number / Nonlinear treatment function / Backward bifurcation / Cost-effectiveness analysis

References

[1] AgustoFB. Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model. BioSystems, 2013, 113: 155-64

[2] ArinoJ, Van den DriesscheP. A multi-city epidemic model. Math Popul Stud, 2003, 10: 175-913

[3] BartlM, LiP, SchusterS. Modelling the optimal timing in metabolic pathway activation-Use of Pontryagin’s Maximum Principle and role of the Golden section. BioSystems, 2010, 101: 67-77

[4] BirkhoffG, RotaCGOrdinary differential equation, 1982BostonGinn and Co.

[5] BuonomoB, LacitignolaD. On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Anal Model Control, 2011, 16: 30-46

[6] BuonomoB, d’OnofrioA, LacitignolaD. Global stability of an SIR epidemic model with information dependent vaccination. Math Biosci, 2008, 216: 9-16

[7] CollinsOC, GovinderKS. Stability analysis and optimal vaccination of a waterborne disease model with multiple water sources. Nat Resour Model, 2016, 29: 426-47

[8] CuiJ, TakeuchiY, SaitoY. Spreading disease with transport-related infection. J Theor Biol, 2006, 239: 376-90

[9] DenphedtnongA, ChinviriyasitS, ChinviriyasitW. On the dynamics of SEIRS epidemic model with transport-related infection. Math Biosci, 2013, 245: 188-205

[10] DiekmannO, HeesterbeekJAPMathematical epidemiology of infectious diseases: model building, analysis and interpretation, 1999New YorkWiley

[11] EckalbarJC, EckalbarWL. Dynamics of an epidemic model with quadratic treatment. Nonlinear Anal Real World Appl, 2011, 12: 320-332

[12] FindlaterA, BogochII. Human mobility and the global spread of infectious diseases: a focus on air travel. Trends Parasitol, 2018, 34: 772-783

[13] JanaS, NandiSK, KarTK. Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment. Acta Biotheor, 2016, 64: 65-84

[14] JanaS, HaldarP, KarTK. Optimal control and stability analysis of an epidemic model with population dispersal. Chaos Solitons Fractals, 2016, 83: 67-81

[15] JanaS, HaldarP, KarTK. Mathematical analysis of an epidemic model with isolation and optimal controls. Int J Comput Math, 2017, 94: 1318-1336

[16] JoshiHR. Optimal control of an HIV immunology model. Optim Control Appl Methods, 2002, 23: 199-213

[17] JungE, LenhartS, FengZ. Optimal control of treatments in a two-strain tuberculosis model. Discrete Contin Dyn Syst Ser B, 2002, 2: 473-482

[18] KarTK, JanaS. A theoretical study on mathematical modelling of an infectious disease with application of optimal control. BioSystems, 2013, 111: 37-50

[19] KarTK, JanaS. Application of three controls optimally in a vector-borne disease—a mathematical study. Commun Nonlinear Sci Numer Simul, 2013, 18: 2868-2884

[20] KarTK, MondalPK. Global dynamics and bifurcation in delayed SIR epidemic model. Nonlinear Anal Real World Appl, 2011, 12: 2058-2068

[21] KarTK, JanaS, GhoraiA. Effect of isolation in an infectious disease. Int J Ecol Econ Stat, 2013, 29: 87-116

[22] KarTK, NandiSK, JanaS, MandalM. Stability and bifurcation analysis of an epidemic model with the effect of media. Chaos Solitons Fractals, 2019, 120: 188-199

[23] KeelingMJ, RohaniPModeling infectious diseases in humans and animals, 2008PrincetonPrinceton University Press

[24] KermackWO, McKendrickAG. Contributions to the mathematical theory of epidemics. Proc R Soc Lond A, 1933, 141: 94-122

[25] KraemerMU, GoldingN, BisanzioD, BhattS, PigottDM, RaySE, BradyOJ, BrownsteinJS, FariaNR, CummingsDA, PybusOG. Utilizing general human movement models to predict the spread of emerging infectious diseases in resource poor settings. Sci Rep, 2019, 9: 1-11

[26] LaarabiH, AbtaA, HattafK. Optimal control of a delayed SIRS epidemic model with vaccination and treatment. Acta Biotheor, 2015, 63: 87-97

[27] LenhartS, WorkmanJTOptimal control applied to biological models, Mathematical and Computational Biology Series, 2007Boca RatonChapman & Hall, CRC Press

[28] LipsitchM, RileyS, CauchemezS, GhaniAC, FergusonNM. Managing and reducing uncertainty in an emerging influenza pandemic. N Engl J Med, 2009, 361: 112-115

[29] LiuX, TakeuchiY. Spread of disease with transport-related infection and entry screening. J Theor Biol, 2006, 242: 517-528

[30] MakindeOD. Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Appl Math Comput, 2007, 184: 842-848

[31] MeloniS, PerraN, ArenasA, GómezS, MorenoY, VespignaniA. Modeling human mobility responses to the large-scale spreading of infectious diseases. Sci Rep, 2011, 1: 62

[32] MisraAK, SharmaA, ShuklaJB. Stability analysis and optimal control of an epidemic model with awareness programs by media. BioSystems, 2015, 138: 53-62

[33] OkosunKO, OuifkiR, MarcusN. Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. BioSystems, 2011, 106: 136-145

[34] OkosunKO, RachidO, MarcusN. Optimal control strategies and cost-effectiveness analysis of a malaria model. BioSystems, 2013, 111: 83-101

[35] PontryaginLS, BoltyanskiiVG, GamkrelidzeRV, MishchenkoEFThe mathematical theory of optimal processes, 1962New YorkWiley

[36] SallahK, GiorgiR, BengtssonL, LuX, WetterE, AdrienP, RebaudetS, PiarrouxR, GaudartJ. Mathematical models for predicting human mobility in the context of infectious disease spread: introducing the impedance model. Int J Health Geogr, 2017, 16: 42

[37] SmithRModelling disease ecology with mathematics, 2008San JoseAmerican Institute of Mathematical Sciences

[38] SunC, YangW, ArinoJ, KhanK. Effect of media-induced social distancing on disease transmission in a two patch setting. Math Biosci, 2011, 230: 87-95

[39] TchuencheJM, KhamisSA, AgustoFB, MpesheSC. Optimal control and sensitivity analysis of an influenza model with treatment and vaccination. Acta Biotheor, 2011, 59: 1-28

[40] ThomaseyDH, MartchevaM. Serotype replacement of vertically transmitted diseases through perfect vaccination. J Biol Syst, 2008, 16: 255-277

[41] Van den DriesscheP, WatmoughJ. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci, 2002, 180: 29-48

[42] WanH, CuiJ. An SEIS epidemic model with transport related infection. J Theor Biol, 2007, 247: 507-524

[43] WangW. Backward bifurcation of an epidemic model with treatment. Math Biosci, 2006, 201: 58-71

[44] WangW, MuloneG. Threshold of disease transmission in a patch environment. J Math Anal Appl, 2003, 285: 321-335

[45] WangW, ZhaoXQ. An epidemic model in a patchy environment. Math Biosci, 2004, 190: 97-112

[46] WangW, ZhaoXQ. An age-structured epidemic model in a patchy environment. SIAM J Appl Math, 2005, 65: 1597-1614

[47] WesolowskiA, BuckeeCO, Engø-MonsenK, MetcalfCJ. Connecting mobility to infectious diseases: the promise and limits of mobile phone data. J Infect Dis, 2016, 214suppl4S414-420

[48] ZhangX, LiuX. Backward bifurcation of an epidemic model with saturated treatment function. J Math Anal Appl, 2008, 348: 433-443

[49] ZhouY, YangK, ZhouK, LiangY. Optimal vaccination policies for an SIR model with limited resources. Acta Biotheor, 2014, 62: 171-181

Funding

Department of Science and Technology-INSPIRE, Government of India(DST/INSPIRE Fellowship/2016/IF160667 dated 21 September 2016)

WBDSTBT(201(Sanc)/S&T/P/ST/16G-12/2018 dated 19/02/2019)

RIGHTS & PERMISSIONS

The Joint Center on Global Change and Earth System Science of the University of Maryland and Beijing Normal University