American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - B, 2022, 27 (2) : 619-638. doi: 10.3934/dcdsb.2021058
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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E. A. Innes, The host-parasite relationship in pregnant cattle infected with Neospora caninum, Parasitology, 134 (2007), 1903-1910.  doi: 10.1017/S0031182007000194.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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L. S. Pontryagin, Mathematical Theory of Optimal Processes, London: Routledge, 1987., doi: 10.1201/9780203749319.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. A. Innes, The host-parasite relationship in pregnant cattle infected with Neospora caninum, Parasitology, 134 (2007), 1903-1910.  doi: 10.1017/S0031182007000194.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25] D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, 1982.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

L. S. Pontryagin, Mathematical Theory of Optimal Processes, London: Routledge, 1987., doi: 10.1201/9780203749319.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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} $
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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 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
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Table 1">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
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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) $
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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) $
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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]
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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
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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
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