# American Institute of Mathematical Sciences

April  2018, 15(2): 543-567. doi: 10.3934/mbe.2018025

## Transmission dynamics and optimal control of brucellosis in Inner Mongolia of China

 1 Department of Applied Mathematics, School of Science, Changchun University of Science and Technology, Changchun 130022, China 2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China 3 School of Mechatronic Engineering, North University of China, Taiyuan 030051, China 4 Complex Systems Research Center, Shanxi University Taiyuan 030006, China 5 China Animal Health And Epidemiology Center, Qingdao 266032, China

* Corresponding author: Meng Fan

Received  November 28, 2016 Accepted  March 15, 2017 Published  June 2017

A multigroup model is developed to characterize brucellosis transmission, to explore potential effects of key factors, and to prioritize control measures. The global threshold dynamics are completely characterized by theory of asymptotic autonomous systems and Lyapunov direct method. We then formulate a multi-objective optimization problem and, by the weighted sum method, transform it into a scalar optimization problem on minimizing the total cost for control. The existence of optimal control and its characterization are well established by Pontryagin's Maximum Principle. We further parameterize the model and compute optimal control strategy for Inner Mongolia in China. In particular, we expound the effects of sheep recruitment, vaccination of sheep, culling of infected sheep, and health education of human on the dynamics and control of brucellosis. This study indicates that current control measures in Inner Mongolia are not working well and Brucellosis will continue to increase. The main finding here supports opposing unregulated sheep breeding and suggests vaccination and health education as the preferred necessary emergency intervention control. The policymakers must take a new look at the current control strategy, and, in order to control brucellosis better in Inner Mongolia, the governments have to preemptively press ahead with more effective measures.

Citation: Linhua Zhou, Meng Fan, Qiang Hou, Zhen Jin, Xiangdong Sun. Transmission dynamics and optimal control of brucellosis in Inner Mongolia of China. Mathematical Biosciences & Engineering, 2018, 15 (2) : 543-567. doi: 10.3934/mbe.2018025
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Schematic transmission diagram of brucellosis among sheep and two human subpopulations. $S, I, V$ represent susceptible, infectious and vaccinated sheep. $S_1$, $I_1$, $C_1$ and $S_2$, $I_2$, $C_2$ represent susceptible, acute cases and chronic cases of high risk human subpopulation $G_1$ and low risk human subpopulation $G_2$, respectively
Predicted tendency of brucellosis in Inner Mongolia. The solid line represents the prediction of (1) and the diamonds are the reported data in Inner Mongolia of P. R. China. (a) Number of brucellosis sheep ($I$). (b) Total number of new confirmed human cases ($I_h^{NA}=I_1^{NA}+I_2^{NA}$). (c) Number of new confirmed human cases of high risk subpopulation $G_1$ ($I_1^{NA}$). (d) Number of new confirmed human cases of low risk subpopulation $G_2$ ($I_2^{NA}$). Here, the culling rate $k=0.05$ from 2001 to 2011 [41], $k=0.5$ from 2012 to 2015 [17,35,40], $k=0.15$ after 2015, $\delta=0.67$, $\theta=0.6$, $q_1=q_2=0.4$, other parameter values are listed in Table 1
Influence of vaccination ($\theta$). The vaccination of sheep can reduce the epidemic situation of brucellosis but can not eliminate it. The values of parameters are the same as those in Fig. 2 except $\theta$
Influence of the recruitment of sheep ($A$). It shows that reducing the size of sheep population in Inner Mongolia could assert great positive effect on brucellosis control. The values of parameters are the same as those in Fig. 2 except $A$
Influence of culling of the infected sheep ($k$). Increasing the culling rate of infected sheep could effectively reduce the incidence of brucellosis in both sheep and human and can even eradicate the brucellosis. The values of parameters are the same as those in Fig. 2 except $k$
Influence of the transmission rate between brucellosis sheep and human($\beta$). Compared to Fig. 4 and Fig. 5, it illustrates that reducing the brucellosis transmission rate between sheep and human has lower efficacy than the controls carried out in sheep stock, since it can not control the source of disease transmission. Here, $\beta_0=4.895\times 10^{-6}$, other values of parameters are the same as those in Fig. 2
The optimal control $\theta^*(t)$, $\varphi_1^*(t)$, $\varphi_2^*(t)$ and the corresponding $I^*(t)$, $I_1^*(t)$, $I_2^*(t)$. The dot-dash curves ($\omega_1=\omega_2=0, \omega_3=\omega_4=0.5$) depict the trends of brucellosis without vaccination programme in sheep stock and health education in human. The dash curves ($\omega_1=\omega_2=0.5, \omega_3=\omega_4=0$) illustrate that the disease can not be eliminated eventually even if the control resources supply is maximal. The solid curves ($\omega_1=\omega_2=\omega_3=0.3,\omega_4=0.1$) show that the situation of brucellosis epidemic will be serious if the control resources supply is limited. Here, $A=3300$, $k=0.15$, $T=34$, $\bar{\theta}=0.85$, $\bar{\varphi_1}=\bar{\varphi_2}=0.9$, and $B_0=0.2$, $B_1=0.3$, $B_2=0.2$, $D_0=1$, $D_1=5$, $D_2=3$, other parameter values are the same as those in Fig. 2
Simulation of the optimal control $\theta^*(t)$, $\varphi_1^*(t)$, $\varphi_2^*(t)$ with varying weighted coefficients. It depicts the significant effect of the weighted coefficients on the optimal control. When $\omega_1$ increase from $0$ to $0.5$, the key minimizing targets switch from vaccination and health education to culling of brucellosis sheep and treatment of human cases, and, accordingly, the optimal control gradually increases from $\theta^*(t)=\varphi_1^*(t)=\varphi_2^*(t)=0$ to $\theta^*(t)=\overline{\theta}$, $\varphi_1^*(t)=\overline{\varphi}_1$ and $\varphi_2^*(t)=\overline{\varphi}_2$. Here $\omega_1=\omega_2, \omega_3=\omega_4$, and $\omega_1+\omega_3=0.5$, other parameter values are the same as those in Fig. 7
Total economic loss (i.e., the sum of the four integrals in (12)) and prevalence rates vary with the weighted coefficients. The policy-maker should confirm the weight coefficients according to the financial budget for brucellosis ($TELoss^*$) and the prevalence rates in both sheep and human population ($P_s^*,P_{h1}^*,P_{h2}^*$). Here $\omega_1=\omega_2, \omega_3=\omega_4$, and $\omega_1+\omega_3=0.5$, other parameter values are the same as those in Fig. 7
Simulations of the optimal control $\theta^*(t)$, $\varphi_1^*(t)$, $\varphi_2^*(t)$ and the corresponding $I^*(t)$, $I_1^*(t)$, $I_2^*(t)$, with the culling rate $k=0.15$, $k=0.3$ and $k=0.7$, respectively. It reveals the fact again that culling of infected sheep plays an important role on brucellosis control. Here, $\omega_1=\omega_2=\omega_3=0.3, \omega_4=0.1$, other parameter values are the same as those in Fig. 7
Simulations of the optimal control $\theta^*(t)$, $\varphi_1^*(t)$, $\varphi_2^*(t)$ and the corresponding $I^*(t)$, $I_1^*(t)$, $I_2^*(t)$ with constant recruitment rate of sheep $A=3300$, $A=2200$ and $A=1600$, respectively. Compared to Fig. 7, the brucellosis epidemic situations are effectively controlled in both sheep and human when the constant recruitment rate of sheep $A$ is decreased from $A=3300$ to $A=1600$. It illustrates that reducing size of sheep population in Inner Mongolia is also an effective control measure. Here, $\omega_1=\omega_2=\omega_3=0.3, \omega_4=0.1$, other parameter values are the same as those in Fig. 7
Parameters of model (1) with default values used for numerical studies
 Parameter Value/Range Unit Definition Reference $A$ 3300 $10^4/$year Constant recruitment of sheep [8] $m$ 0.6 year$^{-1}$ Natural elimination or death rate of sheep [8] $1/\delta$ [1, 3] year Mean effective period of vaccination [9,33,42] $k$ $0-50\%$ year$^{-1}$ Culling rate of infectious sheep [17,35,40,41] $\theta$ [0, 0.85] year$^{-1}$ Effective vaccination rate of susceptible sheep [9,42,33] $\Lambda_1$ 12 $10^4$/year Recruitment of $G_1$ (high risk) [27] $\Lambda_2$ 11 $10^4$/year Recruitment of $G_2$ (low risk) [27] $d_i$ 0.006 year$^{-1}$ Natural death rate of human [27] $1/\gamma_i$ 0.25 year Acute onset period of human [41] $q_i$ [0.32, 0.74] year$^{-1}$ Fraction of acute human cases turned into chronic cases [41] $\lambda$ $2.55\times 10^{-4}$ year$^{-1}$ Transmission rate of sheep Fitting $\beta$ $4.895\times 10^{-6}$ year$^{-1}$ Transmission rate between sheep and $G_1$ Fitting $\varepsilon$ 0.17 year$^{-1}$ Infection risk attenuation coefficient of $G_2$ compared to $G_1$ Fitting Note: $i=1,2$
 Parameter Value/Range Unit Definition Reference $A$ 3300 $10^4/$year Constant recruitment of sheep [8] $m$ 0.6 year$^{-1}$ Natural elimination or death rate of sheep [8] $1/\delta$ [1, 3] year Mean effective period of vaccination [9,33,42] $k$ $0-50\%$ year$^{-1}$ Culling rate of infectious sheep [17,35,40,41] $\theta$ [0, 0.85] year$^{-1}$ Effective vaccination rate of susceptible sheep [9,42,33] $\Lambda_1$ 12 $10^4$/year Recruitment of $G_1$ (high risk) [27] $\Lambda_2$ 11 $10^4$/year Recruitment of $G_2$ (low risk) [27] $d_i$ 0.006 year$^{-1}$ Natural death rate of human [27] $1/\gamma_i$ 0.25 year Acute onset period of human [41] $q_i$ [0.32, 0.74] year$^{-1}$ Fraction of acute human cases turned into chronic cases [41] $\lambda$ $2.55\times 10^{-4}$ year$^{-1}$ Transmission rate of sheep Fitting $\beta$ $4.895\times 10^{-6}$ year$^{-1}$ Transmission rate between sheep and $G_1$ Fitting $\varepsilon$ 0.17 year$^{-1}$ Infection risk attenuation coefficient of $G_2$ compared to $G_1$ Fitting Note: $i=1,2$
Sheep population in Inner Mongolia (unit: 104)
 Year 2001 2002 2003 2004 2005 Sheep Breeding[8] 3551.6 3515.9 3951.7 4450.6 5318.48 Sale and Slaughter[8] 2081.2 2146.5 2156 2867.74 3782.99 Brucellosis sheep1 33.74 35.15 59.27 89.1 132.95 Year 2006 2007 2008 2009 2010 Sheep Breeding[8] 5419.99 5594.44 5063.29 5125.3 5197.2 Sale and Slaughter[8] 4539.6 5011.05 4874.94 5183.7 5339.2 Brucellosis sheep1 162.57 195.79 202.52 230.63 259.85 1 Calculated from data of sheep breeding and the annual seroprevalence of sheep brucella in key monitoring regions of Inner Mongolia[11,13,22,30,38,39,43].
 Year 2001 2002 2003 2004 2005 Sheep Breeding[8] 3551.6 3515.9 3951.7 4450.6 5318.48 Sale and Slaughter[8] 2081.2 2146.5 2156 2867.74 3782.99 Brucellosis sheep1 33.74 35.15 59.27 89.1 132.95 Year 2006 2007 2008 2009 2010 Sheep Breeding[8] 5419.99 5594.44 5063.29 5125.3 5197.2 Sale and Slaughter[8] 4539.6 5011.05 4874.94 5183.7 5339.2 Brucellosis sheep1 162.57 195.79 202.52 230.63 259.85 1 Calculated from data of sheep breeding and the annual seroprevalence of sheep brucella in key monitoring regions of Inner Mongolia[11,13,22,30,38,39,43].
Annual new confirmed cases of human brucellosis in Inner Mongolia[25,41]
 Year 2001 2002 2003 2004 2005 2006 2007 Reported Human Cases 420 610 1280 4140 8740 8050 8117 Year 2008 2009 2010 2011 2012 2013 2014 Reported Human Cases 11105 16551 16935 20845 12817 9310 10538
 Year 2001 2002 2003 2004 2005 2006 2007 Reported Human Cases 420 610 1280 4140 8740 8050 8117 Year 2008 2009 2010 2011 2012 2013 2014 Reported Human Cases 11105 16551 16935 20845 12817 9310 10538
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