Abstract
Foot-and-mouth disease is one of the major contagious zoonotic diseases in the world. It is caused by various species of the genus Aphthovirus of the family Picornavirus, and it always brings a large number of infections and heavy financial losses. The disease has become a major public health concern. In this paper, we propose a nonlocal foot-and-mouth disease model in a spatially heterogeneous environment, which couples virus-to-animals and animals-to-animals transmission pathways, and investigate the dynamics of the disperal. The basic reproduction number \({{\cal R}_0}\) is defined as the spectral radius of the next generation operator \({\cal R}\left( x \right)\) by a renewal equation. The relationship between \({{\cal R}_0}\) and a principal eigenvalue of an operator \({{\cal L}_0}\) is built. Moreover, the proposed system exhibits threshold dynamics in terms of \({{\cal R}_0}\), in the sense that \({{\cal R}_0}\) determines whether or not foot-and-mouth disease invades the hosts. Through numerical simulations, we have found that increasing animals’ movements is an effective control measure for preventing prevalence of the disease.
Similar content being viewed by others
References
Carpenter T E, O’Brien J M, Hagerman A D, McCarl B A. Epidemic and economic impacts of delayed detection of foot-and-mouth disease: a case study of a simulated outbreak in California. J Vet Diagn Invest, 2011, 23: 26–33
Matthews K. A Review of Australias Preparedness for the Threat of Foot-and-Mouth Disease[M/OL]. Canberra, ACT: Australian Government Department of Agriculture, Fisheries and Forestry 2011[2020-03-20]. http://www.agriculture.gov.au/animal-plant-health/pests-diseases-weeds/animal/fmd/review-foot-and-mouth-disease
Rushton J, Knight-Jones T J D, Donaldson A I, deLeeuw P W, Ferrari G, Domenech J. The Impact of Foot and Mouth Disease-Supporting Document N1. Paper prepared for the FAO/OIE Global Conference on Foot and Mouth Disease Control. Thailand: Bangkok, 2012
Brito B P, Rodriguez L L, Hammond J M, Pinto J, Perez A M. Review of the global distribution of foot-andmouth disease virus from 2007 to 2014[J/OL]. Transbound Emerg Dis, 2017, 64: 316–332. https://doi.org/10.1111/tbed.12373
Jamal S M, Belsham G J. Foot-and-mouth disease: past, present and future[J/OL]. Vet Res, 2013, 44: 116. https://doi.org/10.1186/1297-9716-44-116
Meyer R F, Knudsen R C. Foot-and-mouth disease: are view of the virus and the symptoms. J Environ Health, 2001, 64: 21–23
Zhang T L, Zhao X Q. Mathematical modelling for schistosomiasis with seasonal influence: a case study in China. SIAM J Appl Dynam Sys, 2020, 19(2): 1438–1417
Luo X F, Jin Z. A new insight into isolating the high-degree nodes in network to control infectious diseases. Commun Nonlinear Sci Numer Simul, 2020, 91: 105363
Duan X C, Li X Z, Martcheva M. Qualitative analysis on a diffusive age-structured heroin transmission model. Nonlinear Anal: RWA, 2020, 54: 103105
Li X Z, Yang J Y, Martcheva M. Age structured epidemic modelling. Switzerland AG: Springer, 2002
Sellers R F, Gloster J. The Northumberland epidemic of foot-and-mouth disease. 1966. J Hyg (Lond), 1980, 85(1): 129–140
Mikkelsen T, Alexandersen S, Astrup P, et al. Investigation of airborne foot and mouth disease virus transmission during low-wind conditions in the early phase of the UK 2001 epidemic. Atmos Chem Phys, 2003, 3: 2101–2110
Kao R R. The role of mathematical modelling in the control of the 2001 FMD epidemic in the UK. Trends Microbiol, 2002, 10(6): 279–286
Ferguson N M, Donnelly C A, Anderson R M. The foot-and-mouth epidemic in Great Britain: pattern of spread and impact of interventions. Science, 2001, 292: 1155–1160
Ferguson N M, Donnelly C A, Anderson R M. Transmission intensity and impact of control policies on the foot and mouth epidemic in Great Britain. Nature, 2001, 413: 542–548
Bradhurst R A, Roche S E, East I J, et al. A hybrid modelling approach to simulating foot-andmouth disease outbreaks in Australian livestock. Front Environ Sci, 2015, 3: 17
Keeling M J, et al. Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape. Science, 2001, 294: 813–817
Keeling M J, Woolhouse M E J, May R M, et al. Modelling vaccination strategies against foot and mouth disease. Nature, 2003, 421: 136–142
Orsel K, Bouma A. The effect of foot-and-mouth disease (FMD) vaccination on virus transmission and the significance for the field. Can vet J, 2009, 50: 1059–1063
Orsel K, Dekker A, Bouma A, et al. Vaccination against foot and mouth disease reduces virus transmission in groups of calves. Vaccine, 2005, 23(41): 4887–4894
Tildesley M J, Bessell P R, Keeling M J, Woolhouse M E J. The role of pre-emptive culling in the control of foot-and-mouth disease. Proc Roy Soc Lond, B, Biol Sci, 2009, 276(1671): 3239–3248
Bates P W, Zhao G. Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal. J Math Anal Appl, 2007, 332(1): 428–440
Han B S, Yang Y H. On a predator-prey reaction diffusion model with nonlocal effects. Commun Nonlinear Sci Numer Simulat, 2017, 46: 49–61
Zhao G, Ruan S. Spatial and temporal dynamics of a nonlocal viral infection model. SIAM J Appl Math, 2018, 78(4): 1954–1980
Kuniya T, Wang J. Global dynamics of an SIR epidemic modelwith nonlocal diffusion. Nonlinear Anal: RWA, 2018, 43: 262–282
Bates P W. On some nonlocal evolution equations arising in materials science//Brunner H, Xhao X-Q, Zhou X. Nonlinear Dynamics and Evolution Equations. Fields Institute Communications, 2006, 48: 13–52
Tian H, Ju L, Du Q. A conservative nonlocal convection diffusion model and asymptotically compatible finite difference discretization. Comput Meth Appl Mech Engin, 2017, 320: 46–67
Zhang J, Jin Z, Yuan Y. Assessing the spread of foot and mouth disease in mainland China by dynamical switching model. J Theor Biol, 2019, 460: 209–219
Pazy A. Semigroups of Linear Operators and Application to Partial Differential Equations. New York: Springer-Verlag, 1983
Webb G F. Theory of Nonlinear Age-Dependent Population Dynamics. New York: Marcel Dekker Inc, 1985
Wang X Y, Chen Y M, Yang J Y. Spatial and temporal dynamics of a virus infection model with two nonlocal effects[J/OL]. Complexity, 2019, Art ID5842942. https://doi.org/10.1155/2019/5842942
Hale J K. Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs. Providence: American Mathematical Society, 1998
Yang J, Xu F. The computational approach for the basic reproduction number of epidemic models on complex networks. IEEE Access, 2019, 7: 26474–26479
Yang J Y, Jin Z, Xu F. Threshold dynamics of an age-space structured SIR model on heterogeneous environment. Appl Math Letters, 2019, 96: 68–74
Wang W, Zhao X. Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J Appl Dyn Sys, 2011, 11(4): 1652–1673
Diekmann O, Heesterbeek J A P, Metz J A J. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol, 1990, 28: 365–382
LaSalle J P. Some extensions of Liapunov’s second method. Ire Transactions on Circuit Theory, 1960, 7(4): 520–527
Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev, 1976, 18: 620–709
Smith H L, Zhao X Q. Robust persistence for semidynamical systems. Nonlinear Anal: TMA, 2001, 47(9): 6169–6179
Smith H L, Thieme H R. Dynamical systems and population persistence. Providence: American Mathematical Society, 2011
Walker J A. Dynamical Systems and Evolution Equations: Theory and Applications. New York: Plenum Press, 1980
Chatelin F. The spectral approximation of linear operators with applications to the computation of eigenelements of differential and itegral operators. SIAM Rev, 1981, 23: 495–522
Muroya Y, Enatsu Y, Kuniya T. Global stability for a class of multi-group SIR epidemic models with patches through migration and cross path infection. Acta Mathematica Scientia, 2013, 33B(2): 341–361
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the National Natural Science Foundation of China (12001339, 61573016, 11871316), Shanxi Scholarship Council of China (2015-094), the Natural Science Foundation of Shanxi (201801D121006), and the Shanxi Province Science Foundation for Youths (201901D211413).
Rights and permissions
About this article
Cite this article
Wang, X., Yang, J. Dynamics of a Nonlocal Dispersal Foot-and-Mouth Disease Model in a Spatially Heterogeneous Environment. Acta Math Sci 41, 552–572 (2021). https://doi.org/10.1007/s10473-021-0217-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-021-0217-y