# American Institute of Mathematical Sciences

February  2018, 15(1): 323-335. doi: 10.3934/mbe.2018014

## Optimal time to intervene: The case of measles child immunization

 Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, 84248 Bratislava, Slovakia

Received  September 13, 2016 Revised  February 28, 2017 Published  May 2017

The recent measles outbreaks in US and Germany emphasize the importance of sustaining and increasing vaccination rates. In Slovakia, despite mandatory vaccination scheme, decrease in the vaccination rates against measles has been observed in recent years. Different kinds of intervention at the state level, like a law making vaccination a requirement for school entry or education and advertising seem to be the only strategies to improve vaccination coverage. This study aims to analyze the economic effectiveness of intervention in Slovakia. Using real options techniques we determine the level of vaccination rate at which it is optimal to perform intervention. We represent immunization rate of newborns as a stochastic process and intervention as a one-period jump of this process. Sensitivity analysis shows the importance of early intervention in the population with high initial average vaccination coverage. Furthermore, our numerical results demonstrate that the less certain we are about the future development of the immunization rate of newborns, the more valuable is the option to intervene.

Citation: Zuzana Chladná. Optimal time to intervene: The case of measles child immunization. Mathematical Biosciences & Engineering, 2018, 15 (1) : 323-335. doi: 10.3934/mbe.2018014
##### References:
 [1] J. Arino, C. Bauch, F. Brauer, S. M. Driedger, A. L. Greer, S. M. Moghadas, N. J. Pizzi, B. Sander, A. Tuite and P. Van Den Driessche, Pandemic influenza: Modelling and public health perspectives, Math Biosci Eng, 8 (2011), 1-20. doi: 10.3934/mbe.2011.8.1. [2] C. T. Bauch and D. J. Earn, Vaccination and the theory of games, Proceedings of the National Academy of Sciences of the United States of America, 101 (2004), 13391-13394. doi: 10.1073/pnas.0403823101. [3] R. A. Brealey, S. C. Myers and F. Allen, Corporate Finance, Auflage, New York, 2006. [4] B. Buonomo, A. d'Onofrio and D. Lacitignola, Modeling of pseudo-rational exemption to vaccination for seir diseases, Journal of Mathematical Analysis and Applications, 404 (2013), 385-398. doi: 10.1016/j.jmaa.2013.02.063. [5] Z. Chladná and E. Moltchanova, Incentive to vaccinate: A synthesis of two approaches, Acta Mathematica Universitatis Comenianae, 84 (2015), 283-296. [6] T. E. Copeland, V. Antikarov and T. E. Copeland, Real Options: A Practitioner's Guide, Texere New York, 2001. [7] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty, Princeton university press, 1994. [8] A. d'Onofrio, P. Manfredi and P. Poletti, The interplay of public intervention and private choices in determining the outcome of vaccination programmes, PLoS ONE, 7 (2012), e45653. doi: 10.1371/journal.pone.0045653. [9] Analýza epidemiologickej situácie a činnosti odborov epidemiológie v Slovenskej republike. (In Slovak), 2005-2014, URL http://www.epis.sk/InformacnaCast/Publikacie/VyrocneSpravy.aspx. doi: 10.1371/journal.pone.0045653. [10] H. Hudečková, S. Straka, M. Avdičová and S. Rusnáková, Health and economic benefits of mandatory regular vaccination in Slovakia. IV. Measles, rubella and mumps (In Slovak), Epidemiologie, mikrobiologie, imunologie: Časopis Společnosti pro epidemiologii a mikrobiologii České lékařské společnosti JE Purkyně, 50 (2001), 31-35. [11] J. C. Hull, Options, Futures, and Other Derivatives, Pearson Education India, 2006. [12] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, NJ, 2008. [13] P. Manfredi and A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4614-5474-8. [14] Public Health Act, Slovakia 2007, Public Health Protection Act, Slovakia 2007 (In Slovak: Zákon č. 355/2007 Z. z. o ochrane, podpore a rozvoji verejného zdravia a o zmene a doplnení niektorých zákonov), URL http://www.zakonypreludi.sk/zz/2007-355. [15] A. Shefer, P. Briss, L. Rodewald, R. Bernier, R. Strikas, H. Yusuf, S. Ndiaye, S. Wiliams, M. Pappaioanou and A. R. Hinman, Improving immunization coverage rates: An evidence-based review of the literature, Epidemiologic Reviews, 21 (1999), 96-142. doi: 10.1093/oxfordjournals.epirev.a017992. [16] P. J. Smith, S. G. Humiston, E. K. Marcuse, Z. Zhao, C. G. Dorell, C. Howes and B. Hibbs, Parental delay or refusal of vaccine doses, childhood vaccination coverage at 24 months of age, and the health belief model, Public Health Reports, 126 (2011), p135. doi: 10.1177/00333549111260S215. [17] E. Vynnycky and R. White, An Introduction to Infectious Disease Modelling, Oxford University Press, 2010. [18] WHO 2012, Global measles and rubella strategic plan : 2012-2020. The World Health Organization, URL http://apps.who.int/iris/bitstream/10665/44855/1/9789241503396_eng.pdf. [19] WHO 2015, Monitoring and surveillance. Data, statistics and graphics. The World Health Organization, URL http://www.who.int/immunization/monitoring_surveillance/data/en/.

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##### References:
 [1] J. Arino, C. Bauch, F. Brauer, S. M. Driedger, A. L. Greer, S. M. Moghadas, N. J. Pizzi, B. Sander, A. Tuite and P. Van Den Driessche, Pandemic influenza: Modelling and public health perspectives, Math Biosci Eng, 8 (2011), 1-20. doi: 10.3934/mbe.2011.8.1. [2] C. T. Bauch and D. J. Earn, Vaccination and the theory of games, Proceedings of the National Academy of Sciences of the United States of America, 101 (2004), 13391-13394. doi: 10.1073/pnas.0403823101. [3] R. A. Brealey, S. C. Myers and F. Allen, Corporate Finance, Auflage, New York, 2006. [4] B. Buonomo, A. d'Onofrio and D. Lacitignola, Modeling of pseudo-rational exemption to vaccination for seir diseases, Journal of Mathematical Analysis and Applications, 404 (2013), 385-398. doi: 10.1016/j.jmaa.2013.02.063. [5] Z. Chladná and E. Moltchanova, Incentive to vaccinate: A synthesis of two approaches, Acta Mathematica Universitatis Comenianae, 84 (2015), 283-296. [6] T. E. Copeland, V. Antikarov and T. E. Copeland, Real Options: A Practitioner's Guide, Texere New York, 2001. [7] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty, Princeton university press, 1994. [8] A. d'Onofrio, P. Manfredi and P. Poletti, The interplay of public intervention and private choices in determining the outcome of vaccination programmes, PLoS ONE, 7 (2012), e45653. doi: 10.1371/journal.pone.0045653. [9] Analýza epidemiologickej situácie a činnosti odborov epidemiológie v Slovenskej republike. (In Slovak), 2005-2014, URL http://www.epis.sk/InformacnaCast/Publikacie/VyrocneSpravy.aspx. doi: 10.1371/journal.pone.0045653. [10] H. Hudečková, S. Straka, M. Avdičová and S. Rusnáková, Health and economic benefits of mandatory regular vaccination in Slovakia. IV. Measles, rubella and mumps (In Slovak), Epidemiologie, mikrobiologie, imunologie: Časopis Společnosti pro epidemiologii a mikrobiologii České lékařské společnosti JE Purkyně, 50 (2001), 31-35. [11] J. C. Hull, Options, Futures, and Other Derivatives, Pearson Education India, 2006. [12] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, NJ, 2008. [13] P. Manfredi and A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4614-5474-8. [14] Public Health Act, Slovakia 2007, Public Health Protection Act, Slovakia 2007 (In Slovak: Zákon č. 355/2007 Z. z. o ochrane, podpore a rozvoji verejného zdravia a o zmene a doplnení niektorých zákonov), URL http://www.zakonypreludi.sk/zz/2007-355. [15] A. Shefer, P. Briss, L. Rodewald, R. Bernier, R. Strikas, H. Yusuf, S. Ndiaye, S. Wiliams, M. Pappaioanou and A. R. Hinman, Improving immunization coverage rates: An evidence-based review of the literature, Epidemiologic Reviews, 21 (1999), 96-142. doi: 10.1093/oxfordjournals.epirev.a017992. [16] P. J. Smith, S. G. Humiston, E. K. Marcuse, Z. Zhao, C. G. Dorell, C. Howes and B. Hibbs, Parental delay or refusal of vaccine doses, childhood vaccination coverage at 24 months of age, and the health belief model, Public Health Reports, 126 (2011), p135. doi: 10.1177/00333549111260S215. [17] E. Vynnycky and R. White, An Introduction to Infectious Disease Modelling, Oxford University Press, 2010. [18] WHO 2012, Global measles and rubella strategic plan : 2012-2020. The World Health Organization, URL http://apps.who.int/iris/bitstream/10665/44855/1/9789241503396_eng.pdf. [19] WHO 2015, Monitoring and surveillance. Data, statistics and graphics. The World Health Organization, URL http://www.who.int/immunization/monitoring_surveillance/data/en/.
Measles in Slovakia: number of cases and immunization rate of newborns. Source: [19] and [9].
Time development of the average vaccination coverage under different evolution scenarios of immunization rate of newborns.
Expected costs as function of initial average vaccination coverage (left) and initial immunization rate of newborns (right), respectively.
Optimal decisions for different levels of intervention costs: grey = do intervene, black = do not intervene, $t=10$, $\sigma = 0.057$.
Value of the option to intervene. Initial immunization rate of newborns $90\%$ and initial average vaccination coverage $95\%$.
Model inputs.
 Time horizon $T=20$ years Population size $N=400000$ Reproduction number $R_0=17$ Recovery rate $\gamma=0.1$ Birth/death rate $\mu=0.01$ Expected cases $a_0=0.9789$$N, a_1=1.04 Costs per measles case c_0=600€ Intervention costs P\in \langle 0,500000 \rangle € Probability of potential outbreak p_{epi}=0.1 Drift term of immunization rate of newborns \alpha = 0.0076 Volatility of immunization rate of newborns \sigma = 0.057 Discount rate r=1\%  Time horizon T=20 years Population size N=400000 Reproduction number R_0=17 Recovery rate \gamma=0.1 Birth/death rate \mu=0.01 Expected cases a_0=0.9789$$ N$, $a_1=1.04$ Costs per measles case $c_0=600$€ Intervention costs $P\in \langle 0,500000 \rangle$€ Probability of potential outbreak $p_{epi}=0.1$ Drift term of immunization rate of newborns $\alpha = 0.0076$ Volatility of immunization rate of newborns $\sigma = 0.057$ Discount rate $r=1\%$
Savings induced by intervention.
 intervention costs(in thousands of €) savings in exp. costs inital average vacc coverage savings in expected costs 0 8.8% 92% 2.3% 2 6.7% 93% 2.9% 4 4.0% 94% 4.4% 8 3.5% 95% 55.9% 10 2.3% 96% 72.5% 20 2.0% 97% 78.6% 30 1.8% 98% 79.9% (A) Initial immunization rate of newborns $90\%$, initial average vaccination coverage $92\%$, $\sigma =0.057$. (B) Initial immunization rate of newborns $90\%$, initial coverage 10000€, $\sigma =0.057$.
 intervention costs(in thousands of €) savings in exp. costs inital average vacc coverage savings in expected costs 0 8.8% 92% 2.3% 2 6.7% 93% 2.9% 4 4.0% 94% 4.4% 8 3.5% 95% 55.9% 10 2.3% 96% 72.5% 20 2.0% 97% 78.6% 30 1.8% 98% 79.9% (A) Initial immunization rate of newborns $90\%$, initial average vaccination coverage $92\%$, $\sigma =0.057$. (B) Initial immunization rate of newborns $90\%$, initial coverage 10000€, $\sigma =0.057$.
Expected time of first intervention and the expected number of interventions. Intervention costs $P=10000$€, initial average vaccination coverage $\bar x=95\%$ and $\sigma=0.057$.
 Initial immunization rate of newborns Time of first intervention Number of interventionss expected s.e. expected s.e. $90\%$ 6.87 3.79 3.48 3.05 $91\%$ 6.89 3.86 3.20 2.79 $92\%$ 7.92 4.26 3.36 2.81 $93\%$ 7.92 4.34 3.10 2.68 $94\%$ 7.96 4.44 2.89 2.51 $95\%$ 9.19 4.13 3.19 2.71
 Initial immunization rate of newborns Time of first intervention Number of interventionss expected s.e. expected s.e. $90\%$ 6.87 3.79 3.48 3.05 $91\%$ 6.89 3.86 3.20 2.79 $92\%$ 7.92 4.26 3.36 2.81 $93\%$ 7.92 4.34 3.10 2.68 $94\%$ 7.96 4.44 2.89 2.51 $95\%$ 9.19 4.13 3.19 2.71
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