2013, 10(5&6): 1615-1634. doi: 10.3934/mbe.2013.10.1615

Optimal strategies of social distancing and vaccination against seasonal influenza

1. 

Department of Mathematics, University of Tulsa, Tulsa, OK 74104, United States

Received  October 2012 Revised  March 2013 Published  August 2013

Optimal control strategies for controlling seasonal influenza transmission in the US are of high interest, because of the significant epidemiological and economic burden of influenza. To evaluate optimal strategies of vaccination and social distancing, we used an age-structured dynamic model of seasonal influenza. We applied optimal control theory to identify the best way of reducing morbidity and mortality at a minimal cost. In combination with the Pontryagins maximum principle, we calculated time-dependent optimal policies of vaccination and social distancing to minimize the epidemiological and economic burden associated with seasonal influenza. We computed optimal age-specific intervention strategies and analyze them under various costs of interventions and disease transmissibility. Our results show that combined strategies have a stronger impact on the reduction of the final epidemic size. Our results also suggest that the optimal vaccination can be achieved by allocating most vaccines to preschool-age children (age under five) followed by young adults (age 20-39) and school age children (age 6-19). We find that the optimal vaccination rates for all age groups are highest at the beginning of the outbreak, requiring intense effort at the early phase of an epidemic. On the other hand, optimal social distancing of clinical cases tends to last the entire duration of an outbreak, and its intensity is relatively equal for all age groups. Furthermore, with higher transmissibility of the influenza virus (i.e. higher R0), the optimal control strategy needs to include more efforts to increase vaccination rates rather than efforts to encourage social distancing. Taken together, public health agencies need to consider both the transmissibility of the virus and ways to encourage early vaccination as well as voluntary social distancing of symptomatic cases in order to determine optimal intervention strategies against seasonal influenza.
Citation: Eunha Shim. Optimal strategies of social distancing and vaccination against seasonal influenza. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1615-1634. doi: 10.3934/mbe.2013.10.1615
References:
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show all references

References:
[1]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,'', Oxford University Press, (1991). doi: 10.1016/0046-8177(90)90224-S. Google Scholar

[2]

H. Behnke, Optimal control of deterministic epidemics,, Optimal Control Application Methods, 21 (2000), 269. doi: 10.1002/oca.678. Google Scholar

[3]

K. J. Bolton, J. M. McCaw, R. Moss, R. S. Morris, S. Wang, A. Burma, B. Darma, D. Narangerel, P. Nymadawa and J. McVernon, Likely effectiveness of pharmaceutical and non-pharmaceutical interventions for mitigating influenza virus transmission in Mongolia,, Bull World Health Organization, 90 (2012), 264. Google Scholar

[4]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,'', $2^{nd}$ edition, 40 (2012). doi: 10.1007/978-1-4614-1686-9. Google Scholar

[5]

J. S. Brownstein, K. P. Kleinman and K. D. Mandl, Identifying pediatric age groups for influenza vaccination using a real-time regional surveillance system,, Am. J. Epidemiol, 162 (2005), 1. doi: 10.1093/aje/kwi257. Google Scholar

[6]

K. M. Clements, J. Chancellor, K. Nichol, K. DeLong and D. Thompson, Cost-effectiveness of a recommendation of universal mass vaccination for seasonal influenza in the United States,, Value Health, 14 (2011), 800. doi: 10.1016/j.jval.2011.03.005. Google Scholar

[7]

G. Chowell, C. Viboud, X. Wang, S. M. Bertozzi and M. A. Miller, Adaptive vaccination strategies to mitigate pandemic influenza: Mexico as a case study,, PLoS One, 4 (2009). doi: 10.1371/journal.pone.0008164. Google Scholar

[8]

O. Diekmann and J. Heesterbeek, "Mathematical Epidmeiology of Infectious Diseases: Model Building, Analysis and Interpretation,'', Wiley Series in Mathematical and Computational Biology, (2000). Google Scholar

[9]

H. P. Duerr, S. O. Brockmann, I. Piechotowski, M. Schwehm and M. Eichner, Influenza pandemic intervention planning using InfluSim: Pharmaceutical and non-pharmaceutical interventions,, BMC Infect. Dis., 7 (2007). doi: 10.1186/1471-2334-7-76. Google Scholar

[10]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,'', Applications of Mathematics, (1975). Google Scholar

[11]

A. P. Galvani, T. C. Reluga and G. B. Chapman, Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum,, Proc. Natl. Acad. Sci. USA, 104 (2007), 5692. doi: 10.1073/pnas.0606774104. Google Scholar

[12]

R. Gasparini, D. Amicizia, P. L. Lai and D. Panatto, Clinical and socioeconomic impact of seasonal and pandemic influenza in adults and the elderly,, Hum. Vaccin. Immunother., 8 (2012), 21. doi: 10.4161/hv.8.1.17622. Google Scholar

[13]

R. J. Glass, L. M. Glass, W. E. Beyeler and H. J. Min, Targeted social distancing design for pandemic influenza,, Emerg. Infect. Dis., 12 (2006), 1671. doi: 10.3201/eid1211.060255. Google Scholar

[14]

J. Glasser, Z. Feng, A. Moylan, S. Del Valle and C. Castillo-Chavez, Mixing in age-structured population models of infectious diseases,, Math. Biosci., 235 (2012), 1. doi: 10.1016/j.mbs.2011.10.001. Google Scholar

[15]

J. Glasser, D. Taneri, Z. Feng, J. H. Chuang, P. Tll, W. Thompson, M. McCauley and J. Alexander, Evaluation of targeted influenza vaccination strategies via population modeling,, PLoS One, 5 (2010). doi: 10.1371/journal.pone.0012777. Google Scholar

[16]

P. A. Gonzalez-Parra, S. Lee, L. Velazquez and C. Castillo-Chavez, A note on the use of optimal control on a discrete time model of influenza dynamics,, Math. Biosci. Eng., 8 (2011), 183. doi: 10.3934/mbe.2011.8.183. Google Scholar

[17]

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[18]

P. A. Gross, A. W. Hermogenes, H. S. Sacks, J. Lau and R. A. Levandowski, The efficacy of influenza vaccine in elderly persons. A meta-analysis and review of the literature,, Ann. Intern. Med., 123 (1995), 518. doi: 10.7326/0003-4819-123-7-199510010-00008. Google Scholar

[19]

N. Halder, J. K. Kelso and G. J. Milne, Cost-effective strategies for mitigating a future influenza pandemic with H1N1 2009 characteristics,, PLoS One, 6 (2011). doi: 10.1371/journal.pone.0022087. Google Scholar

[20]

J. S. Horvath, M. McKinnon and L. Roberts, The Australian response: Pandemic influenza preparedness,, Med. J. Aust., 185 (2006), 35. Google Scholar

[21]

M. J. Keeling and L. Danon, Mathematical modelling of infectious diseases,, Br. Med. Bull., 92 (2009), 33. doi: 10.1093/bmb/ldp038. Google Scholar

[22]

M. J. Keeling and P. J. White, Targeting vaccination against novel infections: Risk, age and spatial structure for pandemic influenza in Great Britain,, J. R. Soc. Interface, 8 (2011), 661. doi: 10.1098/rsif.2010.0474. Google Scholar

[23]

J. K. Kelso, G. J. Milne and H. Kelly, Simulation suggests that rapid activation of social distancing can arrest epidemic development due to a novel strain of influenza,, BMC Public Health, 9 (2009). doi: 10.1186/1471-2458-9-117. Google Scholar

[24]

S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control for pandemic influenza: The role of limited antiviral treatment and isolation,, J. Theor. Biol., 265 (2010), 136. doi: 10.1016/j.jtbi.2010.04.003. Google Scholar

[25]

S. Lee, M. Golinski and G. Chowell, Modeling optimal age-specific vaccination strategies against pandemic influenza,, Bull. Math. Biol., 74 (2012), 958. doi: 10.1007/s11538-011-9704-y. Google Scholar

[26]

S. Lee, R. Morales and C. Castillo-Chavez, A note on the use of influenza vaccination strategies when supply is limited,, Math. Biosci. Eng., 8 (2011), 171. doi: 10.3934/mbe.2011.8.171. Google Scholar

[27]

V. J. Lee, D. C. Lye and A. Wilder-Smith, Combination strategies for pandemic influenza response - a systematic review of mathematical modeling studies,, BMC Med., 7 (2009). doi: 10.1186/1741-7015-7-76. Google Scholar

[28]

S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models,'', Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007). Google Scholar

[29]

F. Lin, K. Muthuraman and M. Lawley, An optimal control theory approach to non-pharmaceutical interventions,, BMC Infect. Dis., 10 (2010). doi: 10.1186/1471-2334-10-32. Google Scholar

[30]

I. M. Longini, Jr. and M. E. Halloran, Strategy for distribution of influenza vaccine to high-risk groups and children,, Am. J. Epidemiol., 161 (2005), 303. Google Scholar

[31]

A. M. McBean and P. L. Hebert, New estimates of influenza-related pneumonia and influenza hospitalizations among the elderly,, Int. J. Infect. Dis., 8 (2004), 227. doi: 10.1016/j.ijid.2004.04.013. Google Scholar

[32]

J. Medlock and A. P. Galvani, Optimizing influenza vaccine distribution,, Science, 325 (2009), 1705. doi: 10.1126/science.1175570. Google Scholar

[33]

G. N. Mercer, S. I. Barry and H. Kelly, Modelling the effect of seasonal influenza vaccination on the risk of pandemic influenza infection,, BMC Public Health, 11 (2011). doi: 10.1186/1471-2458-11-S1-S11. Google Scholar

[34]

S. M. Moghadas, C. S. Bowman, G. Rst and J. Wu, Population-wide emergence of antiviral resistance during pandemic influenza,, PLoS One, 3 (2008). doi: 10.1371/journal.pone.0001839. Google Scholar

[35]

N. A. Molinari, I. R. Ortega-Sanchez, M. L. Messonnier, W. W. Thompson, P. M. Wortley, E. Weintraub and C. B. Bridges, The annual impact of seasonal influenza in the US: Measuring disease burden and costs,, Vaccine, 25 (2007), 5086. doi: 10.1016/j.vaccine.2007.03.046. Google Scholar

[36]

S. D. Mylius, T. J. Hagenaars, A. K. Lugnr and J. Wallinga, Optimal allocation of pandemic influenza vaccine depends on age, risk and timing,, Vaccine, 26 (2008), 3742. doi: 10.1016/j.vaccine.2008.04.043. Google Scholar

[37]

H. Nishiura, C. Castillo-Chavez, M. Safan and G. Chowell, Transmission potential of the new influenza A(H1N1) virus and its age-specificity in Japan,, Euro. Surveill., 14 (2009). Google Scholar

[38]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,'', Interscience Publishers John Wiley & Sons, (1962). Google Scholar

[39]

O. Prosper, O. Saucedo, D. Thompson, G. Torres-Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza,, Math. Biosci. Eng., 8 (2011), 141. doi: 10.3934/mbe.2011.8.141. Google Scholar

[40]

E. Shim, L. A. Meyers and A. P. Galvani, Optimal H1N1 vaccination strategies based on self-interest versus group interest,, BMC Public Health, 11 (2011). doi: 10.1186/1471-2458-11-S1-S4. Google Scholar

[41]

L. Simonsen, T. A. Reichert, C. Viboud, W. C. Blackwelder, R. J. Taylor and M. A. Miller, Impact of influenza vaccination on seasonal mortality in the US elderly population,, Arch. Intern. Med., 165 (2005), 265. Google Scholar

[42]

J. M. Tchuenche, S. A. Khamis, F. B. Agusto and S. C. Mpeshe, Optimal control and sensitivity analysis of an influenza model with treatment and vaccination,, Acta Biotheor, 59 (2011), 1. doi: 10.1007/s10441-010-9095-8. Google Scholar

[43]

W. W. Thompson, D. K. Shay, E. Weintraub, L. Brammer, C. B. Bridges, N. J. Cox and K. Fukuda, Influenza-associated hospitalizations in the United States,, JAMA, 292 (2004), 1333. doi: 10.1001/jama.292.11.1333. Google Scholar

[44]

W. W. Thompson, D. K. Shay, E. Weintraub, L. Brammer, N. Cox, L. J. Anderson and K. Fukuda, Mortality associated with influenza and respiratory syncytial virus in the United States,, JAMA, 289 (2003), 179. doi: 10.1001/jama.289.2.179. Google Scholar

[45]

S. Towers and Z. Feng, Social contact patterns and control strategies for influenza in the elderly,, Math. Biosci., 240 (2012), 241. doi: 10.1016/j.mbs.2012.07.007. Google Scholar

[46]

J. Truscott, C. Fraser, S. Cauchemez, A. Meeyai, W. Hinsley, C. A. Donnelly, A. Ghani and N. Ferguson, Essential epidemiological mechanisms underpinning the transmission dynamics of seasonal influenza,, J. R. Soc. Interface, 9 (2012), 304. doi: 10.1098/rsif.2011.0309. Google Scholar

[47]

A. R. Tuite, D. N. Fisman, J. C. Kwong and A. L. Greer, Optimal pandemic influenza vaccine allocation strategies for the Canadian population,, PLoS One, 5 (2010). doi: 10.1371/journal.pone.0010520. Google Scholar

[48]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[49]

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