# American Institute of Mathematical Sciences

June  2017, 14(3): 695-708. doi: 10.3934/mbe.2017039

## Mixed vaccination strategy for the control of tuberculosis: A case study in China

 a. College of Mathematics, Jilin University, Changchun 130012, China b. School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China c. Department of Foundation, Aviation University of Air Force, Changchun 130022, China

* Corresponding author: Qingdao Huang

Received  May 13, 2016 Accepted  October 09, 2016 Published  December 2016

Fund Project: The authors are supported by the National Natural Science Foundation of China (11171131)

This study first presents a mathematical model of TB transmission considering BCG vaccination compartment to investigate the transmission dynamics nowadays. Based on data reported by the National Bureau of Statistics of China, the basic reproduction number is estimated approximately as $\mathcal{R}_{0}=1.1892$. To reach the new End TB goal raised by WHO in 2015, considering the health system in China, we design a mixed vaccination strategy. Theoretical analysis indicates that the infectious population asymptotically tends to zero with the new vaccination strategy which is the combination of constant vaccination and pulse vaccination. We obtain that the control of TB is quicker to achieve with the mixed vaccination. The new strategy can make the best of current constant vaccination, and the periodic routine health examination provides an operable environment for implementing pulse vaccination in China. Numerical simulations are provided to illustrate the theoretical results and help to design the final mixed vaccination strategy once the new vaccine comes out.

Citation: Siyu Liu, Yong Li, Yingjie Bi, Qingdao Huang. Mixed vaccination strategy for the control of tuberculosis: A case study in China. Mathematical Biosciences & Engineering, 2017, 14 (3) : 695-708. doi: 10.3934/mbe.2017039
##### References:
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Wang, A Systematic review on the protective efficacy of BCG against children tuberculosis meningitis and millet tuberculosis, Chinese Journal of Vaccines and Immunization, 15 (2009), 359-362. Google Scholar [32] B. Xu, Y. Hu, Q. Zhao, W. Wang, W. Jiang and G. Zhao, Molecular epidemiology of TB -Its impact on multidrug-resistant tuberculosis control in China, International Journal of Mycobacteriology, 4 (2015), 134. doi: 10.1016/j.ijmyco.2014.09.003. Google Scholar [33] Y. Yang, S. Tang, X. Ren, H. Zhao and C. Guo, Global stability and optimal control for a tuberculosis model with vaccination and treatment, Discrete and Continuous Dynamical Systems -Series B, 21 (2016), 1009-1022. doi: 10.3934/dcdsb.2016.21.1009. Google Scholar [34] Y. Zhou, K. Khan, Z. Feng and J. Wu, Projection of tuberculosis incidence with increasing immigration trends, Journal of Theoretical Biology, 254 (2008), 215-228. doi: 10.1016/j.jtbi.2008.05.026. Google Scholar [35] E. Ziv, C. L. Daley and S. M. Blower, Early therapy for latent tuberculosis infection, American Journal of Epidemiology, 153 (2001), 381-385. doi: 10.1093/aje/153.4.381. Google Scholar

show all references

##### References:
 [1] J. P. Aparicio and C. Castillo-Chavez, Mathematical modelling of tuberculosis epidemics, Mathematical Biosciences and Engineering, 6 (2009), 209-237. doi: 10.3934/mbe.2009.6.209. Google Scholar [2] Z. Bai, Threshold dynamics of a time-delayed SEIRS model with pulse vaccination, Mathematical Biosciences, 269 (2015), 178-185. doi: 10.1016/j.mbs.2015.09.005. Google Scholar [3] S. M. Blower and T. Chou, Modeling the emergence of the 'hot zones': Tuberculosis and the amplification dynamics of drug resistance, Nature Medicine, 10 (2004), 1111-1116. doi: 10.1038/nm1102. Google Scholar [4] S. M. Blower, P. M. Small and P. C. Hopewell, Control strategies for tuberculosis epidemic: New models for old problems, Science, 273 (1996), 497-500. doi: 10.1126/science.273.5274.497. Google Scholar [5] S. Bowong and J. J. Tewa, Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 3621-3631. doi: 10.1016/j.cnsns.2010.01.007. Google Scholar [6] H. Cao and Y. Zhou, The discrete age-structured SEIT model with application to tuberculosis transmission in China, Mathematical and Computer Modelling, 55 (2012), 385-395. doi: 10.1016/j.mcm.2011.08.017. Google Scholar [7] C. Castillo-Chavez and Z. Feng, To treat or not treat: The case of tuberculosis, Journal of Mathematical Biology, 35 (1997), 629-656. doi: 10.1007/s002850050069. Google Scholar [8] C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Mathematical Biosciences, 151 (1998), 135-154. Google Scholar [9] 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. Google Scholar [10] H. Chang, Quality monitoring and effect evaluation of BCG vaccination in neonatus, Occupation and Health, 9 (2013), 1109-1110. Google Scholar [11] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_{0}$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324. Google Scholar [12] P. V. D. 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. Google Scholar [13] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008. Google Scholar [14] J. Li, The spread and prevention of tuberculosis, Chinese Remedies and Clinics, 13 (2013), 482-483. Google Scholar [15] L. Liu, X. Zhao and Y. Zhou, A tuberculosis model with seasonality, Bulletin of Mathematical Biology, 72 (2010), 931-952. doi: 10.1007/s11538-009-9477-8. Google Scholar [16] J. S. Lopes, P. Rodrigues, S. T. Pinho, R. F. Andrade, R. Duarte and M. G. M. Gomes, Interpreting Measures of Tuberculosis Transmission: A Case Study on the Portuguese Population, BMC Infectious Diseases, 2014. doi: 10.1186/1471-2334-14-340. Google Scholar [17] Z. Lu, X. Chi and L. Chen, The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Mathematical and Computer Modelling, 36 (2002), 1039-1057. doi: 10.1016/S0895-7177(02)00257-1. Google Scholar [18] National Bureau of Statistics of China, Statistical Data of Tuberculosis 2004-2014. Available from: http://data.stats.gov.cn/easyquery.htm?cn=C01&zb=A0O0F01.Google Scholar [19] National Bureau of Statistics of China, China Statistical Yearbook 2014, Birth Rate, Death Rate and Natural Growth Rate of Population, 2014. Available from: http://www.stats.gov.cn/tjsj/ndsj/2014/indexch.htm.Google Scholar [20] National Technic Steering Group of the Epidemiological Sampling Survey for Tuberculosis, Report on fourth national epidemiological sampling survey of tuberculosis, Chinese Journal of Tuberculosis and Respiratory Diseases, 25 (2002), 3-7.Google Scholar [21] A. M. Samoilenko and N. A. Perestyuk, Periodic and almost-periodic solution of impulsive differential equations, Ukrainian Mathematical Journal, 34 (1982), 66-73,132. doi: 10.1007/BF01086134. Google Scholar [22] A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, Visa Skola, Kiev, 1987.Google Scholar [23] O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Mathematical Biosciences and Engineering, 5 (2008), 145-174. doi: 10.3934/mbe.2008.5.145. Google Scholar [24] M. Shen, Y. Xiao, W. Zhou and Z. Li, Global dynamics and applications of an epidemiological model for hepatitis C virus transmission in China, Discrete Dynamics in Nature and Society, 2015 (2015), Article ID 543029, 13pp. doi: 10.1155/2015/543029. Google Scholar [25] B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60 (1998), 1123-1148. Google Scholar [26] B. Song, C. Castillo-Chavez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts, Mathematical Biosciences, 180 (2002), 187-205. doi: 10.1016/S0025-5564(02)00112-8. Google Scholar [27] Technical Guidance Group of the Fifth National TB Epidemiological Survey and The Office of the Fifth National TB Epidemiological Survey, The fifth national tuberculosis epidemiological survey in 2010, Chinese Journal of Antituberculosis, 34 (2012), 485–508.Google Scholar [28] E. Vynnycky and P. E. M. Fine, The long-term dynamics of tuberculosis and other diseases with long serial intervals: Implications of and for changing reproduction numbers, Epidemiology and Infection, 121 (1998), 309-324. doi: 10.1017/S0950268898001113. Google Scholar [29] WHO, Global Tuberculosis Report 2015, World Health Organization, 2015. Available from: http://www.who.int/tb/publications/global_report/en/.Google Scholar [30] WHO, Tuberculosis Vaccine Development, 2015. Available from: http://www.who.int/immunization/research/development/tuberculosis/en/.Google Scholar [31] C. Xiong, X. Liang and H. Wang, A Systematic review on the protective efficacy of BCG against children tuberculosis meningitis and millet tuberculosis, Chinese Journal of Vaccines and Immunization, 15 (2009), 359-362. Google Scholar [32] B. Xu, Y. Hu, Q. Zhao, W. Wang, W. Jiang and G. Zhao, Molecular epidemiology of TB -Its impact on multidrug-resistant tuberculosis control in China, International Journal of Mycobacteriology, 4 (2015), 134. doi: 10.1016/j.ijmyco.2014.09.003. Google Scholar [33] Y. Yang, S. Tang, X. Ren, H. Zhao and C. Guo, Global stability and optimal control for a tuberculosis model with vaccination and treatment, Discrete and Continuous Dynamical Systems -Series B, 21 (2016), 1009-1022. doi: 10.3934/dcdsb.2016.21.1009. Google Scholar [34] Y. Zhou, K. Khan, Z. Feng and J. Wu, Projection of tuberculosis incidence with increasing immigration trends, Journal of Theoretical Biology, 254 (2008), 215-228. doi: 10.1016/j.jtbi.2008.05.026. Google Scholar [35] E. Ziv, C. L. Daley and S. M. Blower, Early therapy for latent tuberculosis infection, American Journal of Epidemiology, 153 (2001), 381-385. doi: 10.1093/aje/153.4.381. Google Scholar
(a)Annual newly reported TB cases in mainland China (b)Goodness of fit and prediction of TB trends until 2024. Asterisks represent the newly reported TB cases by year. The blue solid line shows the fit based on the current circumstances. All the parameters are shown in Table 1
Comparison between the only constant vaccination strategy and mixed vaccination strategy. The black solid line shows the constant vaccination strategy (with $p=0.95$). The blue solid line shows mixed vaccination (with $p=0.7, p_{c}=0.25$ and $T=5$). The red solid line represents the goal. Other parameters are shown in Table 1
Time-series of the susceptible, infectious, exposed and recovered population evolving according to the mixed vaccination SEIR model 8 with $p=0.8, p_{c}=0.3$ and $T=3$. All the other parameters are shown in Table 1
The effect of different starting time on duration of mixed vaccination strategy with $p=0.8, p_{c}=0.3$ and $T=3$. The white bar represents the period of current vaccination strategy and the grey bar shows the duration of mixed vaccination strategy needs to achieve the goal. All the other parameters are shown in Table 1
(a)The effect of the vaccination period of pulse vaccination on the control of TB, set $p=0.8, p_{c}=0.3$ and T varies from $3$ to $6$ years by step-size of $1$ year. (b)The effect of the pulse inoculation rate on the control of TB, set $p=0.8, T=3$ and $p_{c}$ varies from $10\%$ to $40\%$ by step-size of $10\%$. All the other parameters are shown in Table 1
Parameters and initial data
 Parm & Init D. Description Value Source $\Lambda$ Recruitment rate $1.6\times10^{7} \ year^{{-1}}$ see text $d$ Natural death rate $0.0139 \ year^{{-1}}$ [19] $b$ Natural birth rate $0.0123 \ year^{{-1}}$ [19] $\beta$ Transmission rate of infected population $0.5905$ LS $\sigma$ Disease-induced death rate $0.06 \ year^{{-1}}$ [15,35] $\varepsilon$ Rate of progression to infectious stage from the exposed $6 \ year^{{-1}}$ [14] $k$ Rate of waning immunity $0.25 \ year^{{-1}}$ [10,14,31] $\gamma$ Recovery rate $0.4055 \ year^{{-1}}$ LS $p_{b}$ The fraction of BCG vaccinated successfully 0.6 [10,14,31] $B(0)$ Initial number of BCG vaccinated successfully population $3.84\times 10^{7}$ Calculated $S(0)$ Initial number of susceptible population $9.44\times 10^{8}$ LS $E(0)$ Initial number of exposed population $1.62\times 10^{5}$ [18] $I(0)$ Initial number of infected population $3.10\times 10^{6}$ [18,20] $R(0)$ Initial number of recovered population $3.14\times 10^{8}$ LS Parm, Parameter; Init D., Initial Data; LS, least square.
 Parm & Init D. Description Value Source $\Lambda$ Recruitment rate $1.6\times10^{7} \ year^{{-1}}$ see text $d$ Natural death rate $0.0139 \ year^{{-1}}$ [19] $b$ Natural birth rate $0.0123 \ year^{{-1}}$ [19] $\beta$ Transmission rate of infected population $0.5905$ LS $\sigma$ Disease-induced death rate $0.06 \ year^{{-1}}$ [15,35] $\varepsilon$ Rate of progression to infectious stage from the exposed $6 \ year^{{-1}}$ [14] $k$ Rate of waning immunity $0.25 \ year^{{-1}}$ [10,14,31] $\gamma$ Recovery rate $0.4055 \ year^{{-1}}$ LS $p_{b}$ The fraction of BCG vaccinated successfully 0.6 [10,14,31] $B(0)$ Initial number of BCG vaccinated successfully population $3.84\times 10^{7}$ Calculated $S(0)$ Initial number of susceptible population $9.44\times 10^{8}$ LS $E(0)$ Initial number of exposed population $1.62\times 10^{5}$ [18] $I(0)$ Initial number of infected population $3.10\times 10^{6}$ [18,20] $R(0)$ Initial number of recovered population $3.14\times 10^{8}$ LS Parm, Parameter; Init D., Initial Data; LS, least square.
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