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doi: 10.3934/dcdsb.2021058
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## Stability analysis and optimal control of production-limiting disease in farm with two vaccines

 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 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.

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, doi: 10.3934/dcdsb.2021058
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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 $\rho_h = \zeta\frac{I}{N}$
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
">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
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 $p_2(t)$
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 $p_2(t)$
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]
 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]
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
 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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