Modeling the intrinsic dynamics of foot-and-mouth disease

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2016, 13(2): 425-442. doi: 10.3934/mbe.2015010

Modeling the intrinsic dynamics of foot-and-mouth disease

Steady Mushayabasa ^1, , Drew Posny ^2, and Jin Wang ^3,

1.	Department of Mathematics, University of Zimbabwe, P.O. Box MP 167, Harare, Zimbabwe
2.	NSF Center for Integrated Pest Management, NC State University, Raleigh, NC 27606, United States
3.	Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States

Received April 2015 Revised November 2015 Published December 2015

Abstract
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We propose a new mathematical modeling framework to investigate the transmission and spread of foot-and-mouth disease. Our models incorporate relevant biological and ecological factors, vaccination effects, and seasonal impacts during the complex interaction among susceptible, vaccinated, exposed, infected, carrier, and recovered animals. We conduct both epidemic and endemic analysis, with a focus on the threshold dynamics characterized by the basic reproduction numbers. In addition, numerical simulation results are presented to demonstrate the analytical findings.

Keywords: threshold dynamics., mathematical models, Foot-and-mouth disease.

Mathematics Subject Classification: 92B05, 93A30, 93C1.

Citation: Steady Mushayabasa, Drew Posny, Jin Wang. Modeling the intrinsic dynamics of foot-and-mouth disease. Mathematical Biosciences & Engineering, 2016, 13 (2) : 425-442. doi: 10.3934/mbe.2015010

References:

[1]	F. Aftosa, Foot and Mouth Disease, http://www.cfsph.iastate.edu/Factsheets/pdfs/foot_and_mouth_disease.pdf, (Accessed September 2014)
[2]	A. Alexandersen, Z. Zhang and I. A. Donaldson, Aspects of the persistence of foot-and -mouth disease virus in animals-the carrier problem, Microbes and Infection, 4 (2002), 1099-1110.
[3]	C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.
[4]	M. E. Chase-Topping, I. Handel, B. M. Bankowski, N. D. Juleff, D. Gibson, S. J. Cox, M. A. Windsor, E. Reid, C. Doel, R. Howey, P. V. Barnett, M. E. J. Woolhouse and B. Charleston, Understanding foot-and-mouth disease virus transmission biology: Identification of the indicators of infectiousness, Veterinary Research, 44 (2013), p46. doi: 10.1186/1297-9716-44-46.
[5]	T. A. Dekker, H. Vernooij, A. Bouma and A. Stegeman, Rate of foot-and-mouth disease virus transmission by carriers quantified from experimental data, Risk Analysis, 28 (2008), 303-309. doi: 10.1111/j.1539-6924.2008.01020.x.
[6]	O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, 2000.
[7]	J. Gloster, H. Champion, J. Sorensen, T. Mikkelsen, D. Ryall, P. Astrup, S. Alexandersen and A. Donaldson, Airborne transmission of foot-and-mouth disease virus from Burnside Farm, Heddon-on-the-Wall, Northumberland, during the 2001 epidemic in the United Kingdom, The Veterinary Record, 152 (2003), 525-533.
[8]	R. R. Kao, L. Danon, D. M. Green and I. Z. Kiss, Demographic structure and pathogen dynamics on the network of livestock movements in Great Britain, Proceedings of the Royal Society B, 273 (2006), 1999-2007. doi: 10.1098/rspb.2006.3505.
[9]	M. J. Keeling, M. E. J. Woolhouse, R. M. May, G. Davies and B. T. Grenfell, Modelling vaccination strategies against foot-and-mouth disease, Nature, 421 (2003), 136-142. doi: 10.1038/nature01343.
[10]	R. P. Kitching, Identification of foot and mouth disease virus carrier and subclinically infected animals and differentiation from vaccinated animals, Revue scientifique et technique (International Office of Epizootics), 21 (2002), 531-538.
[11]	R. P. Kitching, A. M. Humber and M. V. Thrusfield, A review of foot-and-mouth disease with special consideration for the clinical and epidemiological factors relevant to predictive modelling of the disease, Veterinary Journal, 169 (2005), 197-209. doi: 10.1016/j.tvjl.2004.06.001.
[12]	T. J. Knight-Jones, A. N. Bulut, K. D. Stark, D. U. Pfeiffer, K. J. Sumption and D. J. Paton, Retrospective evaluation of foot-and-mouth disease vaccine effectiveness in Turkey, Vaccine, 32 (2014), 1848-1855. doi: 10.1016/j.vaccine.2014.01.071.
[13]	G. E. Lahodny Jr., R. Gautam and R. Ivanek, Estimating the probability of an extinction or major outbreak for an environmentally transmitted infectious disease, Journal of Biological Dynamics, 9 (2015), 128-155. doi: 10.1080/17513758.2014.954763.
[14]	J. S. LaSalle, The stability of Dynamical Systems, SIAM: Philadelphia, 1976.
[15]	S. Mushayabasa, C. P. Bhunu and M. Dhlamini, Impact of vaccination and culling on controlling foot and mouth disease: a mathematical modeling approach, World Journal of Vaccines, 1 (2011), 156-161. doi: 10.4236/wjv.2011.14016.
[16]	S. Parida, Vaccination against foot-and-mouth disease virus: Strategies and effectiveness, Expert Review of Vaccines, 8 (2009), 347-365. doi: 10.1586/14760584.8.3.347.
[17]	D. Posny and J. Wang, Modelling cholera in periodic environments, Journal of Biological Dynamics, 8 (2014), 1-19. doi: 10.1080/17513758.2014.896482.
[18]	D. Posny and J. Wang, Computing basic reproductive numbers for epidemiological models in nonhomogeneous environments, Applied Mathematics and Computation, 242 (2014), 473-490. doi: 10.1016/j.amc.2014.05.079.
[19]	N. Ringa and C. T. Bauch, Impacts of constrained culling and vaccination on control of foot and mouth disease in near-endemic settings: A pair approximation model, Epidemics, 9 (2014), 18-30. doi: 10.1016/j.epidem.2014.09.008.
[20]	Y. Sinkala, M. Simuunza, J. B. Muma, D. U. Pfeiffer, C. J. Kasanga and A. Mweene, Foot and mouth disease in Zambia: Spatial and temporal distributions of outbreaks, assessment of clusters and implications for control, Onderstepoort Journal of Veterinary Research, 81 (2014), 6pp. doi: 10.4102/ojvr.v81i2.741.
[21]	P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[22]	W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.
[23]	X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

References:

[1]	F. Aftosa, Foot and Mouth Disease, http://www.cfsph.iastate.edu/Factsheets/pdfs/foot_and_mouth_disease.pdf, (Accessed September 2014)
[2]	A. Alexandersen, Z. Zhang and I. A. Donaldson, Aspects of the persistence of foot-and -mouth disease virus in animals-the carrier problem, Microbes and Infection, 4 (2002), 1099-1110.
[3]	C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.
[4]	M. E. Chase-Topping, I. Handel, B. M. Bankowski, N. D. Juleff, D. Gibson, S. J. Cox, M. A. Windsor, E. Reid, C. Doel, R. Howey, P. V. Barnett, M. E. J. Woolhouse and B. Charleston, Understanding foot-and-mouth disease virus transmission biology: Identification of the indicators of infectiousness, Veterinary Research, 44 (2013), p46. doi: 10.1186/1297-9716-44-46.
[5]	T. A. Dekker, H. Vernooij, A. Bouma and A. Stegeman, Rate of foot-and-mouth disease virus transmission by carriers quantified from experimental data, Risk Analysis, 28 (2008), 303-309. doi: 10.1111/j.1539-6924.2008.01020.x.
[6]	O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, 2000.
[7]	J. Gloster, H. Champion, J. Sorensen, T. Mikkelsen, D. Ryall, P. Astrup, S. Alexandersen and A. Donaldson, Airborne transmission of foot-and-mouth disease virus from Burnside Farm, Heddon-on-the-Wall, Northumberland, during the 2001 epidemic in the United Kingdom, The Veterinary Record, 152 (2003), 525-533.
[8]	R. R. Kao, L. Danon, D. M. Green and I. Z. Kiss, Demographic structure and pathogen dynamics on the network of livestock movements in Great Britain, Proceedings of the Royal Society B, 273 (2006), 1999-2007. doi: 10.1098/rspb.2006.3505.
[9]	M. J. Keeling, M. E. J. Woolhouse, R. M. May, G. Davies and B. T. Grenfell, Modelling vaccination strategies against foot-and-mouth disease, Nature, 421 (2003), 136-142. doi: 10.1038/nature01343.
[10]	R. P. Kitching, Identification of foot and mouth disease virus carrier and subclinically infected animals and differentiation from vaccinated animals, Revue scientifique et technique (International Office of Epizootics), 21 (2002), 531-538.
[11]	R. P. Kitching, A. M. Humber and M. V. Thrusfield, A review of foot-and-mouth disease with special consideration for the clinical and epidemiological factors relevant to predictive modelling of the disease, Veterinary Journal, 169 (2005), 197-209. doi: 10.1016/j.tvjl.2004.06.001.
[12]	T. J. Knight-Jones, A. N. Bulut, K. D. Stark, D. U. Pfeiffer, K. J. Sumption and D. J. Paton, Retrospective evaluation of foot-and-mouth disease vaccine effectiveness in Turkey, Vaccine, 32 (2014), 1848-1855. doi: 10.1016/j.vaccine.2014.01.071.
[13]	G. E. Lahodny Jr., R. Gautam and R. Ivanek, Estimating the probability of an extinction or major outbreak for an environmentally transmitted infectious disease, Journal of Biological Dynamics, 9 (2015), 128-155. doi: 10.1080/17513758.2014.954763.
[14]	J. S. LaSalle, The stability of Dynamical Systems, SIAM: Philadelphia, 1976.
[15]	S. Mushayabasa, C. P. Bhunu and M. Dhlamini, Impact of vaccination and culling on controlling foot and mouth disease: a mathematical modeling approach, World Journal of Vaccines, 1 (2011), 156-161. doi: 10.4236/wjv.2011.14016.
[16]	S. Parida, Vaccination against foot-and-mouth disease virus: Strategies and effectiveness, Expert Review of Vaccines, 8 (2009), 347-365. doi: 10.1586/14760584.8.3.347.
[17]	D. Posny and J. Wang, Modelling cholera in periodic environments, Journal of Biological Dynamics, 8 (2014), 1-19. doi: 10.1080/17513758.2014.896482.
[18]	D. Posny and J. Wang, Computing basic reproductive numbers for epidemiological models in nonhomogeneous environments, Applied Mathematics and Computation, 242 (2014), 473-490. doi: 10.1016/j.amc.2014.05.079.
[19]	N. Ringa and C. T. Bauch, Impacts of constrained culling and vaccination on control of foot and mouth disease in near-endemic settings: A pair approximation model, Epidemics, 9 (2014), 18-30. doi: 10.1016/j.epidem.2014.09.008.
[20]	Y. Sinkala, M. Simuunza, J. B. Muma, D. U. Pfeiffer, C. J. Kasanga and A. Mweene, Foot and mouth disease in Zambia: Spatial and temporal distributions of outbreaks, assessment of clusters and implications for control, Onderstepoort Journal of Veterinary Research, 81 (2014), 6pp. doi: 10.4102/ojvr.v81i2.741.
[21]	P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[22]	W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.
[23]	X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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