Stability analysis and optimal control of production-limiting disease in farm with two vaccines

Yue Liu; Wing-Cheong Lo

doi:10.3934/dcdsb.2021058

Previous Article
Qualitative analysis of integro-differential equations with variable retardation
DCDS-B Home
This Issue
Next Article

February 2022, 27(2): 619-638. doi: 10.3934/dcdsb.2021058

Stability analysis and optimal control of production-limiting disease in farm with two vaccines

Yue Liu ^, and Wing-Cheong Lo

Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China

^* Corresponding author: Yue Liu

Received September 2020 Revised December 2020 Published February 2022 Early access February 2021

Fund Project: This work was supported by CityU Strategic Research Grants (Project Nos. CityU 11303719 and CityU 11301520)

The transmission of production-limiting disease in farm, such as Neosporosis and Johne's disease, has brought a huge loss worldwide due to reproductive failure. This paper aims to provide a modeling framework for controlling the disease and investigating the spread dynamics of Neospora caninum-infected dairy as a case study. In particular, a dynamic model for production-limiting disease transmission in the farm is proposed. It incorporates the vertical and horizontal transmission routes and two vaccines. The threshold parameter, basic reproduction number $ \mathcal{R}_0 $, is derived and qualitatively used to explore the stability of the equilibria. Global stability of the disease-free and endemic equilibria is investigated using the comparison theorem or geometric approach. On the case study of Neospora caninum-infected dairy in Switzerland, sensitivity analysis of all involved parameters with respect to the basic reproduction number $ \mathcal{R}_0 $ has been performed. Through Pontryagin's maximum principle, the optimal control problem is discussed to determine the optimal vaccination coverage rate while minimizing the number of infected individuals and control cost at the same time. Moreover, numerical simulations are performed to support the analytical findings. The present study provides useful information on the understanding of production-limiting disease prevention on a farm.

Keywords: Production-limiting disease, reproduction number, global stability, optimal control theory, geometric approach.

Mathematics Subject Classification: Primary: 34D20, 34D23; Secondary: 37N25, 92B05.

Citation: Yue Liu, Wing-Cheong Lo. Stability analysis and optimal control of production-limiting disease in farm with two vaccines. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 619-638. doi: 10.3934/dcdsb.2021058

References:

[1]	G. S. Abdellrazeq, M. M. Elnaggar, J. P. Bannantine, D. A. Schneider, C. D. Souza, J. Hwang, A. H. A. Mahmoud, V. Hulubei, L. M. Fry, K.-T. Park and W. C. Davis, A peptide-based vaccine for Mycobacterium avium subspecies paratuberculosis, Vaccine, 37 (2019), 2783-2790. doi: 10.1016/j.vaccine.2019.04.040.
[2]	R. A. Atkinson, R. W. Cook, L. A. Reddacliff, J. Rothwell, K. W. Broady, P. Harper and J. T. Ellis, Seroprevalence of Neospora caninum infection following an abortion outbreak in a dairy cattle herd, Aust. Vet. J., 78 (2000), 262-266. doi: 10.1111/j.1751-0813.2000.tb11752.x.
[3]	I. A. Baba, R. A. Abdulkadir and P. Esmaili, Analysis of tuberculosis model with saturated incidence rate and optimal control, Physica A, 540 (2020), 123237. doi: 10.1016/j.physa.2019.123237.
[4]	C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of R0 and its role on global stability, In: Mathematical approaches for emerging and reemerging infectious disease: An introduction, 125, Springer, New York, 2002,229–250., doi: 10.1007/978-1-4757-3667-0_13.
[5]	K. Chakraborty, S. Jana and T. K. Kar, Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting, Appl. Math. Comput., 218 (2012), 9271-9290. doi: 10.1016/j.amc.2012.03.005.
[6]	H. C. Davison, C. S. Guy, J. W. McGarry, F. Guy, D. Williams, D. F. Kelly and A. J. Trees, Experimental studies on the transmission of Neospora caninum between cattle, Res. Vet. Sci., 70 (2001), 163-168.
[7]	H. C. Davison, A. Otter and A. J. Trees, Estimation of vertical and horizontal transmission parameters of Neospora caninum infections in dairy cattle, Int. J. Parasitol., 29 (1999), 1683-1689. doi: 10.1016/S0020-7519(99)00129-0.
[8]	T. Dijkstra, H. W. Barkema, J. W. Hesselink and W. Wouda, Point source exposure of cattle to Neospora caninum consistent with periods of common housing and feeding and related to the introduction of a dog, Vet. Parasitol., 105 (2002), 89-98. doi: 10.1016/S0304-4017(02)00009-2.
[9]	J. P. Dubey, Review of Neospora caninum and neosporosis in animals, Korean J. Parasitol., 41 (2003), 1-16. doi: 10.3347/kjp.2003.41.1.1.
[10]	N. P. French, D. Clancy, H. C. Davison and A. J. Trees, Mathematical models of Neospora caninum infection in dairy cattle: transmission and options for control, Int. J. Parasitol., 29 (1999), 1691-1704. doi: 10.1016/S0020-7519(99)00131-9.
[11]	T. B. Gashirai, S. D. Musekwa-Hove, P. O. Lolika and S. Mushayabasa, Global stability and optimal control analysis of a foot-and-mouth disease model with vaccine failure and environmental transmission, Chaos Solit. Fract., 132 (2020), 109568. doi: 10.1016/j.chaos.2019.109568.
[12]	H. Groenendaal, M. Nielen and J. W. Hesselink, Development of the Dutch Johne's disease control program supported by a simulation model, Prev. Vet. Med., 60 (2003), 69-90. doi: 10.1016/S0167-5877(03)00083-7.
[13]	C. A. Hall, M. P. Reichel and J. T. Ellis, Neospora abortions in dairy cattle: diagnosis, mode of transmission and control, Vet. Parasitol., 128 (2005), 231-241. doi: 10.1016/j.vetpar.2004.12.012.
[14]	B. Häsler, K. D. C. Stärk, H. Sager, B. Gottstein and M. Reist, Simulating the impact of four control strategies on the population dynamics of Neospora caninum infection in Swiss dairy cattle, Prev. Vet. Med., 77 (2006), 254-283. doi: 10.1016/j.prevetmed.2006.07.007.
[15]	J. Hernandez, C. Risco and A. Donovan, Association between exposure to Neospora caninum and milk production in dairy cows, J. Am. Vet. Med. Assoc., 219 (2001), 632-635. doi: 10.2460/javma.2001.219.632.
[16]	L. Huo, L. Wang and X. Zhao, Stability analysis and optimal control of a rumor spreading model with media report, Physica A, 517 (2019), 551-562. doi: 10.1016/j.physa.2018.11.047.
[17]	E. A. Innes, The host-parasite relationship in pregnant cattle infected with Neospora caninum, Parasitology, 134 (2007), 1903-1910. doi: 10.1017/S0031182007000194.
[18]	E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst.-Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473.
[19]	T. Kang, Q. Zhang and L. Rong, A delayed avian influenza model with avian slaughter: Stability analysis and optimal control, Physica A, 529 (2019), 121544, 16 pp. doi: 10.1016/j.physa.2019.121544.
[20]	T. K. Kar, S. K. Nandi, S. Jana and M. Mandal, Stability and bifurcation analysis of an epidemic model with the effect of media, Chaos Solit. Fract., 120 (2019), 188-199. doi: 10.1016/j.chaos.2019.01.025.
[21]	M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449.
[22]	S. Liddell, M. C. Jenkins, C. M. Collica and J. P. Dubey, Prevention of vertical transfer of Neospora caninum in BALB/c mice by vaccination, J. Parasitol., 85 (1999), 1072-1075. doi: 10.2307/3285670.
[23]	Y. Liu, M. P. Reichel and W.-C. Lo, Combined control evaluation for Neospora caninum infection in dairy: Economic point of view coupled with population dynamics, Vet. Parasitol., 277 (2020), 108967. doi: 10.1016/j.vetpar.2019.108967.
[24]	Z. Lu, R. M. Mitchell, R. L. Smith, J. S. Van Kessel, P. P. Chapagain, Y. H. Schukken and Y. T. Grohn, The importance of culling in Johne's disease control, J. Theor. Biol., 254 (2008), 135-146. doi: 10.1016/j.jtbi.2008.05.008.
[25]	D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, 1982.
[26]	R. H. Martin, Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45 (1974), 432-454. doi: 10.1016/0022-247X(74)90084-5.
[27]	C. Miller, H. Quinn, C. Ryce, M. P. Reichel and J. T. Ellis, Reduction in transplacental transmission of Neospora caninum in outbred mice by vaccination, Int. J. Parasitol., 35 (2005), 821-828. doi: 10.1016/j.ijpara.2005.03.006.
[28]	R. M. Mitchell, R. H. Whitlock, S. M. Stehman, A. Benedictus, P. P. Chapagain, Y. T. Grohn and Y. H. Schukken, Simulation modeling to evaluate the persistence of Mycobacterium avium subsp. paratuberculosis (MAP) on commercial dairy farms in the United States, Prev. Vet. Med., 83 (2008), 360-380. doi: 10.1016/j.prevetmed.2007.09.006.
[29]	A. R. Moen, W. Wouda, M. F. Mul, E. A. M. Graat and T. van Werven, Increased risk of abortion following Neospora caninum abortion outbreaks: A retrospective and prospective cohort study in four dairy herds, Theriogenology, 49 (1998), 1301-1309. doi: 10.1016/S0093-691X(98)00077-6.
[30]	K. Murakami, S. Kobayashi, M. Konishi, K. ichiro Kameyama, T. Yamamoto and T. Tsutsui, The recent prevalence of bovine leukemia virus (BLV) infection among Japanese cattle, Vet. Microbiol., 148 (2011), 84-88. doi: 10.1016/j.vetmic.2010.08.001.
[31]	S. L. Ott, S. J. Wells and B. A. Wagner, Herd-level economic losses associated with Johne's disease on US dairy operations, Prev. Vet. Med., 40 (1999), 179-192. doi: 10.1016/S0167-5877(99)00037-9.
[32]	D. U Pfeiffer, N. B Williamson, M. P Reichel, J. J Wichtel and W. R Teague, A longitudinal study of Neospora caninum infection on a dairy farm in {New Zealand}, Prev. Vet. Med., 54 (2002), 11-24. doi: 10.1016/S0167-5877(02)00011-9.
[33]	L. S. Pontryagin, Mathematical Theory of Optimal Processes, London: Routledge, 1987., doi: 10.1201/9780203749319.
[34]	M. P. Reichel, M. Alejandra Ayanegui-Alcérreca, L. F. P. Gondim and J. T. Ellis, What is the global economic impact of Neospora caninum in cattle-The billion dollar question, Int. J. Parasitol., 43 (2013), 133-142. doi: 10.1016/j.ijpara.2012.10.022.
[35]	M. P. Reichel and J. T. Ellis, If control of Neospora caninum infection is technically feasible does it make economic sense?, Vet. Parasitol., 142 (2006), 23-34. doi: 10.1016/j.vetpar.2006.06.027.
[36]	G. Schares, M. Peters, R. Wurm, A. Bärwald and F. J. Conraths, The efficiency of vertical transmission of Neospora caninum in dairy cattle analysed by serological techniques, Vet. Parasitol., 80 (1998), 87-98. doi: 10.1016/S0304-4017(98)00195-2.
[37]	S. Schärrer, P. Presi, J. Hattendorf, N. Chitnis, M. Reist and J. Zinsstag, Demographic model of the Swiss cattle population for the years 2009-2011 stratified by gender, age and production type, PLoS One, 9 (2014), 1-10. doi: 10.1371/journal.pone.0109329.
[38]	A. J. Trees and D. J. L. Williams, Endogenous and exogenous transplacental infection in Neospora caninum and Toxoplasma gondii, Trends Parasitol., 21 (2005), 558-561. doi: 10.1016/j.pt.2005.09.005.
[39]	D. J. L. Williams and A. J. Trees, Protecting babies: Vaccine strategies to prevent foetopathy in Neospora caninum-infected cattle, Parasite Immunol., 28 (2006), 61-67. doi: 10.1111/j.1365-3024.2005.00809.x.
[40]	D. J. L. Williams, C. S. Guy, R. F. Smith, F. Guy, J. W. McGarry, J. S. McKay and A. J. Trees, First demonstration of protective immunity against foetopathy in cattle with latent Neospora caninum infection, Int. J. Parasitol., 33 (2003), 1059-1065. doi: 10.1016/S0020-7519(03)00143-7.
[41]	D. J. Wilson, K. Orsel, J. Waddington, M. Rajeev, A. R. Sweeny, T. Joseph, M. E. Grigg and S. A. Raverty, Neospora caninum is the leading cause of bovine fetal loss in British Columbia, Canada, Vet. Parasitol., 218 (2016), 46-51. doi: 10.1016/j.vetpar.2016.01.006.
[42]	Y. Yang, Global stability of VEISV propagation modeling for network worm attack, Appl. Math. Model., 39 (2015), 776-780. doi: 10.1016/j.apm.2014.07.010.
[43]	A. Yousefpour, H. Jahanshahi and S. Bekiros, Optimal policies for control of the novel coronavirus disease (COVID-19) outbreak, Chaos Solit. Fract., 136 (2020), 109883, 6 pp. doi: 10.1016/j.chaos.2020.109883.

show all references

References:

[1]	G. S. Abdellrazeq, M. M. Elnaggar, J. P. Bannantine, D. A. Schneider, C. D. Souza, J. Hwang, A. H. A. Mahmoud, V. Hulubei, L. M. Fry, K.-T. Park and W. C. Davis, A peptide-based vaccine for Mycobacterium avium subspecies paratuberculosis, Vaccine, 37 (2019), 2783-2790. doi: 10.1016/j.vaccine.2019.04.040.
[2]	R. A. Atkinson, R. W. Cook, L. A. Reddacliff, J. Rothwell, K. W. Broady, P. Harper and J. T. Ellis, Seroprevalence of Neospora caninum infection following an abortion outbreak in a dairy cattle herd, Aust. Vet. J., 78 (2000), 262-266. doi: 10.1111/j.1751-0813.2000.tb11752.x.
[3]	I. A. Baba, R. A. Abdulkadir and P. Esmaili, Analysis of tuberculosis model with saturated incidence rate and optimal control, Physica A, 540 (2020), 123237. doi: 10.1016/j.physa.2019.123237.
[4]	C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of R0 and its role on global stability, In: Mathematical approaches for emerging and reemerging infectious disease: An introduction, 125, Springer, New York, 2002,229–250., doi: 10.1007/978-1-4757-3667-0_13.
[5]	K. Chakraborty, S. Jana and T. K. Kar, Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting, Appl. Math. Comput., 218 (2012), 9271-9290. doi: 10.1016/j.amc.2012.03.005.
[6]	H. C. Davison, C. S. Guy, J. W. McGarry, F. Guy, D. Williams, D. F. Kelly and A. J. Trees, Experimental studies on the transmission of Neospora caninum between cattle, Res. Vet. Sci., 70 (2001), 163-168.
[7]	H. C. Davison, A. Otter and A. J. Trees, Estimation of vertical and horizontal transmission parameters of Neospora caninum infections in dairy cattle, Int. J. Parasitol., 29 (1999), 1683-1689. doi: 10.1016/S0020-7519(99)00129-0.
[8]	T. Dijkstra, H. W. Barkema, J. W. Hesselink and W. Wouda, Point source exposure of cattle to Neospora caninum consistent with periods of common housing and feeding and related to the introduction of a dog, Vet. Parasitol., 105 (2002), 89-98. doi: 10.1016/S0304-4017(02)00009-2.
[9]	J. P. Dubey, Review of Neospora caninum and neosporosis in animals, Korean J. Parasitol., 41 (2003), 1-16. doi: 10.3347/kjp.2003.41.1.1.
[10]	N. P. French, D. Clancy, H. C. Davison and A. J. Trees, Mathematical models of Neospora caninum infection in dairy cattle: transmission and options for control, Int. J. Parasitol., 29 (1999), 1691-1704. doi: 10.1016/S0020-7519(99)00131-9.
[11]	T. B. Gashirai, S. D. Musekwa-Hove, P. O. Lolika and S. Mushayabasa, Global stability and optimal control analysis of a foot-and-mouth disease model with vaccine failure and environmental transmission, Chaos Solit. Fract., 132 (2020), 109568. doi: 10.1016/j.chaos.2019.109568.
[12]	H. Groenendaal, M. Nielen and J. W. Hesselink, Development of the Dutch Johne's disease control program supported by a simulation model, Prev. Vet. Med., 60 (2003), 69-90. doi: 10.1016/S0167-5877(03)00083-7.
[13]	C. A. Hall, M. P. Reichel and J. T. Ellis, Neospora abortions in dairy cattle: diagnosis, mode of transmission and control, Vet. Parasitol., 128 (2005), 231-241. doi: 10.1016/j.vetpar.2004.12.012.
[14]	B. Häsler, K. D. C. Stärk, H. Sager, B. Gottstein and M. Reist, Simulating the impact of four control strategies on the population dynamics of Neospora caninum infection in Swiss dairy cattle, Prev. Vet. Med., 77 (2006), 254-283. doi: 10.1016/j.prevetmed.2006.07.007.
[15]	J. Hernandez, C. Risco and A. Donovan, Association between exposure to Neospora caninum and milk production in dairy cows, J. Am. Vet. Med. Assoc., 219 (2001), 632-635. doi: 10.2460/javma.2001.219.632.
[16]	L. Huo, L. Wang and X. Zhao, Stability analysis and optimal control of a rumor spreading model with media report, Physica A, 517 (2019), 551-562. doi: 10.1016/j.physa.2018.11.047.
[17]	E. A. Innes, The host-parasite relationship in pregnant cattle infected with Neospora caninum, Parasitology, 134 (2007), 1903-1910. doi: 10.1017/S0031182007000194.
[18]	E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst.-Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473.
[19]	T. Kang, Q. Zhang and L. Rong, A delayed avian influenza model with avian slaughter: Stability analysis and optimal control, Physica A, 529 (2019), 121544, 16 pp. doi: 10.1016/j.physa.2019.121544.
[20]	T. K. Kar, S. K. Nandi, S. Jana and M. Mandal, Stability and bifurcation analysis of an epidemic model with the effect of media, Chaos Solit. Fract., 120 (2019), 188-199. doi: 10.1016/j.chaos.2019.01.025.
[21]	M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449.
[22]	S. Liddell, M. C. Jenkins, C. M. Collica and J. P. Dubey, Prevention of vertical transfer of Neospora caninum in BALB/c mice by vaccination, J. Parasitol., 85 (1999), 1072-1075. doi: 10.2307/3285670.
[23]	Y. Liu, M. P. Reichel and W.-C. Lo, Combined control evaluation for Neospora caninum infection in dairy: Economic point of view coupled with population dynamics, Vet. Parasitol., 277 (2020), 108967. doi: 10.1016/j.vetpar.2019.108967.
[24]	Z. Lu, R. M. Mitchell, R. L. Smith, J. S. Van Kessel, P. P. Chapagain, Y. H. Schukken and Y. T. Grohn, The importance of culling in Johne's disease control, J. Theor. Biol., 254 (2008), 135-146. doi: 10.1016/j.jtbi.2008.05.008.
[25]	D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, 1982.
[26]	R. H. Martin, Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45 (1974), 432-454. doi: 10.1016/0022-247X(74)90084-5.
[27]	C. Miller, H. Quinn, C. Ryce, M. P. Reichel and J. T. Ellis, Reduction in transplacental transmission of Neospora caninum in outbred mice by vaccination, Int. J. Parasitol., 35 (2005), 821-828. doi: 10.1016/j.ijpara.2005.03.006.
[28]	R. M. Mitchell, R. H. Whitlock, S. M. Stehman, A. Benedictus, P. P. Chapagain, Y. T. Grohn and Y. H. Schukken, Simulation modeling to evaluate the persistence of Mycobacterium avium subsp. paratuberculosis (MAP) on commercial dairy farms in the United States, Prev. Vet. Med., 83 (2008), 360-380. doi: 10.1016/j.prevetmed.2007.09.006.
[29]	A. R. Moen, W. Wouda, M. F. Mul, E. A. M. Graat and T. van Werven, Increased risk of abortion following Neospora caninum abortion outbreaks: A retrospective and prospective cohort study in four dairy herds, Theriogenology, 49 (1998), 1301-1309. doi: 10.1016/S0093-691X(98)00077-6.
[30]	K. Murakami, S. Kobayashi, M. Konishi, K. ichiro Kameyama, T. Yamamoto and T. Tsutsui, The recent prevalence of bovine leukemia virus (BLV) infection among Japanese cattle, Vet. Microbiol., 148 (2011), 84-88. doi: 10.1016/j.vetmic.2010.08.001.
[31]	S. L. Ott, S. J. Wells and B. A. Wagner, Herd-level economic losses associated with Johne's disease on US dairy operations, Prev. Vet. Med., 40 (1999), 179-192. doi: 10.1016/S0167-5877(99)00037-9.
[32]	D. U Pfeiffer, N. B Williamson, M. P Reichel, J. J Wichtel and W. R Teague, A longitudinal study of Neospora caninum infection on a dairy farm in {New Zealand}, Prev. Vet. Med., 54 (2002), 11-24. doi: 10.1016/S0167-5877(02)00011-9.
[33]	L. S. Pontryagin, Mathematical Theory of Optimal Processes, London: Routledge, 1987., doi: 10.1201/9780203749319.
[34]	M. P. Reichel, M. Alejandra Ayanegui-Alcérreca, L. F. P. Gondim and J. T. Ellis, What is the global economic impact of Neospora caninum in cattle-The billion dollar question, Int. J. Parasitol., 43 (2013), 133-142. doi: 10.1016/j.ijpara.2012.10.022.
[35]	M. P. Reichel and J. T. Ellis, If control of Neospora caninum infection is technically feasible does it make economic sense?, Vet. Parasitol., 142 (2006), 23-34. doi: 10.1016/j.vetpar.2006.06.027.
[36]	G. Schares, M. Peters, R. Wurm, A. Bärwald and F. J. Conraths, The efficiency of vertical transmission of Neospora caninum in dairy cattle analysed by serological techniques, Vet. Parasitol., 80 (1998), 87-98. doi: 10.1016/S0304-4017(98)00195-2.
[37]	S. Schärrer, P. Presi, J. Hattendorf, N. Chitnis, M. Reist and J. Zinsstag, Demographic model of the Swiss cattle population for the years 2009-2011 stratified by gender, age and production type, PLoS One, 9 (2014), 1-10. doi: 10.1371/journal.pone.0109329.
[38]	A. J. Trees and D. J. L. Williams, Endogenous and exogenous transplacental infection in Neospora caninum and Toxoplasma gondii, Trends Parasitol., 21 (2005), 558-561. doi: 10.1016/j.pt.2005.09.005.
[39]	D. J. L. Williams and A. J. Trees, Protecting babies: Vaccine strategies to prevent foetopathy in Neospora caninum-infected cattle, Parasite Immunol., 28 (2006), 61-67. doi: 10.1111/j.1365-3024.2005.00809.x.
[40]	D. J. L. Williams, C. S. Guy, R. F. Smith, F. Guy, J. W. McGarry, J. S. McKay and A. J. Trees, First demonstration of protective immunity against foetopathy in cattle with latent Neospora caninum infection, Int. J. Parasitol., 33 (2003), 1059-1065. doi: 10.1016/S0020-7519(03)00143-7.
[41]	D. J. Wilson, K. Orsel, J. Waddington, M. Rajeev, A. R. Sweeny, T. Joseph, M. E. Grigg and S. A. Raverty, Neospora caninum is the leading cause of bovine fetal loss in British Columbia, Canada, Vet. Parasitol., 218 (2016), 46-51. doi: 10.1016/j.vetpar.2016.01.006.
[42]	Y. Yang, Global stability of VEISV propagation modeling for network worm attack, Appl. Math. Model., 39 (2015), 776-780. doi: 10.1016/j.apm.2014.07.010.
[43]	A. Yousefpour, H. Jahanshahi and S. Bekiros, Optimal policies for control of the novel coronavirus disease (COVID-19) outbreak, Chaos Solit. Fract., 136 (2020), 109883, 6 pp. doi: 10.1016/j.chaos.2020.109883.

Figure 1. Flow diagram describing the vaccination model in disease-infected dairy cattle. Animals in cattle are categorized into four compartments: susceptible animals ($ S $), infected animals ($ I $), susceptible vaccinees ($ V $), and infected vaccinees ($ W $). The dashed line indicates that the buying or selling rate $ \Lambda $ is a variable and depending on other compartments. Here the horizontal infection rate $ \rho_h = \zeta\frac{I}{N} $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Solutions with different initial values converge to the (a) DFE and (b) EE. (a) All the parameters, except $ \rho_{v1} = 0.5 $, are used as given in Table 1 where the criterion for global asymptotic stability of DFE is satisfied. (b) $ \alpha = 0.34 $, $ \beta_{I1} = 0.05 $, $ \rho_{v2} = 0.5 $, $ \delta = 0.2 $, $ r = 0.2 $, $ p_1 = \phi_1 = \phi_2 = 0.2 $, $ p_2 = 0.1 $ and other parameters not specified are used as given in Table 1. With these parameter values, the criterion for global stability of EE is satisfied

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Uncertainty analysis of the basic reproduction number (a) $ \mathcal{\widehat{R}}_0 $ without control and (b) $ \mathcal{R}_0 $ with vaccination control. Frequency distribution of basic reproduction number $ \mathcal{\widehat{R}}_0 $ and $ \mathcal{R}_0 $. All the parameters are used as Table 1

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Simulation results of optimal control with low control cost $ C_1 = 1 $, $ C_2 = 5 $. (a)–(c) Number of infected individuals ($ I $), susceptible vaccinees ($ V $), and infected vaccinees ($ W $). The dashed and solid lines represent, respectively, the population dynamics with regular control and optimal control. (d) Control profiles of $ p_1(t) $ and $ p_2(t) $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Simulation results of optimal control with high control cost $ C_1 = 20 $, $ C_2 = 100 $. (a)–(c) Number of infected individuals ($ I $), susceptible vaccinees ($ V $), and infected vaccinees ($ W $). The dashed and solid lines represent, respectively, the population dynamics with regular control and optimal control. (d) Control profiles of $ p_1(t) $ and $ p_2(t) $

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. Parameters used in the numerical simulations

Notation	Definition	Unit	Value	Reference
$ \Lambda $	Buying or selling rate	$ \text{Year}^{-1} $	–	–
$ \alpha $	Pregnant rate	$ \text{Year}^{-1} $	0.30	[14,37]
$ \beta_S $	Abortion rate of susceptible pregnant cow	$ \text{Unitless} $	0.02	[14]
$ \beta_{I1} $	Abortion rate of infected pregnant cow	$ \text{Unitless} $	0.08	[29]
$ \beta_{I2} $	Abortion rate of infected vaccinee	$ \text{Unitless} $	0.05	[29]
$ \rho_{v1} $	Vertical infection rate of infected cow	$ \text{Unitless} $	0.95	[13,14]
$ \rho_{v2} $	Vertical infection rate of infected vaccinee	$ \text{Unitless} $	0.40	[13,14]
$ \zeta $	Prevalence dependent factor	$ \text{Year}^{-1} $	0.028	[10,14]
$ p_1 $	Proportion of the vaccine 1	$ \text{Year}^{-1} $	0.50	Assumption
$ p_2 $	Proportion of the vaccine 2	$ \text{Year}^{-1} $	0.50	Assumption
$ \delta $	Mortality rate	$ \text{Year}^{-1} $	0.10	Assumption
$ \epsilon $	Removal rate	$ \text{Year}^{-1} $	0.095	Assumption
$ \phi_1 $	Efficacy of the vaccine 1	Unitless	0.60	Assumption
$ \phi_2 $	Efficacy of the vaccine 2	Unitless	0.60	Assumption
$ r $	Proportion of cows	Unitless	0.50	[23]

Table Options

Download as excel

Notation	Definition	Unit	Value	Reference
$ \Lambda $	Buying or selling rate	$ \text{Year}^{-1} $	–	–
$ \alpha $	Pregnant rate	$ \text{Year}^{-1} $	0.30	[14,37]
$ \beta_S $	Abortion rate of susceptible pregnant cow	$ \text{Unitless} $	0.02	[14]
$ \beta_{I1} $	Abortion rate of infected pregnant cow	$ \text{Unitless} $	0.08	[29]
$ \beta_{I2} $	Abortion rate of infected vaccinee	$ \text{Unitless} $	0.05	[29]
$ \rho_{v1} $	Vertical infection rate of infected cow	$ \text{Unitless} $	0.95	[13,14]
$ \rho_{v2} $	Vertical infection rate of infected vaccinee	$ \text{Unitless} $	0.40	[13,14]
$ \zeta $	Prevalence dependent factor	$ \text{Year}^{-1} $	0.028	[10,14]
$ p_1 $	Proportion of the vaccine 1	$ \text{Year}^{-1} $	0.50	Assumption
$ p_2 $	Proportion of the vaccine 2	$ \text{Year}^{-1} $	0.50	Assumption
$ \delta $	Mortality rate	$ \text{Year}^{-1} $	0.10	Assumption
$ \epsilon $	Removal rate	$ \text{Year}^{-1} $	0.095	Assumption
$ \phi_1 $	Efficacy of the vaccine 1	Unitless	0.60	Assumption
$ \phi_2 $	Efficacy of the vaccine 2	Unitless	0.60	Assumption
$ r $	Proportion of cows	Unitless	0.50	[23]

Table 2. The sensitivity indices of $ \mathcal{\widehat{R}}_0 $ and $ \mathcal{R}_0 $ to the parameter $ v $

Parameter ($ v $)	Value	Sensitivity index of $ \mathcal{\widehat{R}}_0 $	Sensitivity index of $ \mathcal{R}_0 $
$ \alpha $	0.30	0.8240	0.5225
$ \beta_{I1} $	0.08	-0.0717	-0.0240
$ \rho_{v1} $	0.95	0.8240	0.2751
$ \zeta $	0.028	0.1760	0.0059
$ \delta $	0.10	-0.6893	-0.0311
$ \epsilon $	0.095	-0.3103	-0.2203
$ r $	0.50	0.5137	0.3023
$ \beta_{I2} $	0.05	–	-0.0129
$ \rho_{v2} $	0.40	–	0.2475
$ p_1 $	0.50	–	-0.3219
$ p_2 $	0.50	–	-0.4267
$ \phi_1 $	0.60	–	-0.3219
$ \phi_2 $	0.60	–	-0.4267

Table Options

Download as excel

Parameter ($ v $)	Value	Sensitivity index of $ \mathcal{\widehat{R}}_0 $	Sensitivity index of $ \mathcal{R}_0 $
$ \alpha $	0.30	0.8240	0.5225
$ \beta_{I1} $	0.08	-0.0717	-0.0240
$ \rho_{v1} $	0.95	0.8240	0.2751
$ \zeta $	0.028	0.1760	0.0059
$ \delta $	0.10	-0.6893	-0.0311
$ \epsilon $	0.095	-0.3103	-0.2203
$ r $	0.50	0.5137	0.3023
$ \beta_{I2} $	0.05	–	-0.0129
$ \rho_{v2} $	0.40	–	0.2475
$ p_1 $	0.50	–	-0.3219
$ p_2 $	0.50	–	-0.4267
$ \phi_1 $	0.60	–	-0.3219
$ \phi_2 $	0.60	–	-0.4267

[1]	K. Renee Fister, Jennifer Hughes Donnelly. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences & Engineering, 2005, 2 (3) : 499-510. doi: 10.3934/mbe.2005.2.499
[2]	Elie Assémat, Marc Lapert, Dominique Sugny, Steffen J. Glaser. On the application of geometric optimal control theory to Nuclear Magnetic Resonance. Mathematical Control and Related Fields, 2013, 3 (4) : 375-396. doi: 10.3934/mcrf.2013.3.375
[3]	Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial and Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73
[4]	María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics and Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004
[5]	Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. A minimal approach to the theory of global attractors. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2079-2088. doi: 10.3934/dcds.2012.32.2079
[6]	Jing-Jing Xiang, Juan Wang, Li-Ming Cai. Global stability of the dengue disease transmission models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2217-2232. doi: 10.3934/dcdsb.2015.20.2217
[7]	Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009
[8]	B. Bonnard, J.-B. Caillau, E. Trélat. Geometric optimal control of elliptic Keplerian orbits. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 929-956. doi: 10.3934/dcdsb.2005.5.929
[9]	Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063
[10]	Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107
[11]	Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415
[12]	Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3157-3187. doi: 10.3934/dcdss.2020163
[13]	Jorge San Martín, Takéo Takahashi, Marius Tucsnak. An optimal control approach to ciliary locomotion. Mathematical Control and Related Fields, 2016, 6 (2) : 293-334. doi: 10.3934/mcrf.2016005
[14]	Djamila Moulay, M. A. Aziz-Alaoui, Hee-Dae Kwon. Optimal control of chikungunya disease: Larvae reduction, treatment and prevention. Mathematical Biosciences & Engineering, 2012, 9 (2) : 369-392. doi: 10.3934/mbe.2012.9.369
[15]	Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934/mbe.2007.4.633
[16]	Shangbing Ai. Global stability of equilibria in a tick-borne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567-572. doi: 10.3934/mbe.2007.4.567
[17]	C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008
[18]	Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
[19]	Dingjun Yao, Rongming Wang, Lin Xu. Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission. Journal of Industrial and Management Optimization, 2015, 11 (2) : 461-478. doi: 10.3934/jimo.2015.11.461
[20]	Monique Chyba, Geoff Patterson, Gautier Picot, Mikael Granvik, Robert Jedicke, Jeremie Vaubaillon. Designing rendezvous missions with mini-moons using geometric optimal control. Journal of Industrial and Management Optimization, 2014, 10 (2) : 477-501. doi: 10.3934/jimo.2014.10.477

2020 Impact Factor: 1.327

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.327

Tools

Figures and Tables

2020 Impact Factor: 1.327

American Institute of Mathematical Sciences

Stability analysis and optimal control of production-limiting disease in farm with two vaccines

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Stability analysis and optimal control of production-limiting disease in farm with two vaccines

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors