# American Institute of Mathematical Sciences

May  2017, 22(3): 687-715. doi: 10.3934/dcdsb.2017034

## Optimal control and cost-effectiveness analysis of a three age-structured transmission dynamics of chikungunya virus

 Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence KS 66045, USA

Email: fbagusto@gmail.com

Received  August 2015 Revised  December 2015 Published  December 2016

Chikungunya is an RNA viral disease, transmitted to humans by infected Aedes aegypti or Aedes albopictus mosquitoes. In this paper, an age-structured deterministic model for the transmission dynamics of Chikungunya virus is presented. The model is locally and globally asymptotically stable when the reproduction number is less than unity. A global sensitivity analysis using the reproduction number indicates that the mosquito biting rate, the transmission probability per contact of mosquitoes and of humans, mosquito recruitment rate and the death rate of the mosquitoes are the parameters with the most influence on Chikungunya transmission dynamics. Optimal control theory was then applied, using the results from the sensitivity analysis, to minimize the number infected humans, with time dependent control variables (impacting mosquito biting rate, transmission probability, death rate and recovery rates in humans).

The numerical simulations indicate that Chikungunya can be reduced by the application of these controls. The benefits associated with these health interventions are evaluated using cost-effectiveness analysis and these shows that using mono-control strategy involving treatment of infected individuals is the most cost-effective strategy of this category. With pairs of control, the pairs involving treatment of infected individuals and mosquitoes adulticiding, is the most cost-effective strategy of this category and is more cost-effective than using the triple control strategy involving personal protection, treatment of infected humans and mosquitoes adulticiding.

Citation: Folashade B. Agusto. Optimal control and cost-effectiveness analysis of a three age-structured transmission dynamics of chikungunya virus. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 687-715. doi: 10.3934/dcdsb.2017034
##### References:
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##### References:
 [1] F. B. Agusto, S. Easley, K. Freeman and M. Thomas, Mathematical model of a three age-structured transmission dynamics of Chikungunya Virus Comput. Math. Methods Med. , (2016), Art. ID 4320514, 31 pp. Google Scholar [2] F. B. Agusto, Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model, BioSystems, 113 (2013), 155-164.  doi: 10.1016/j.biosystems.2013.06.004.  Google Scholar [3] F. B. Agusto, J. M. Lenhart and S. Cushing, Optimal Control of the spread of malaria super-infectivity, Journal of Biological Systems Special issue on Infectious Disease Modeling, 21 (2013), 1340002, 26pp.  doi: 10.1142/S0218339013400020.  Google Scholar [4] F. B. Agusto, N. Marcus and K. O. Okosun, Application of optimal control to the epidemiology of malaria disease, Electronic Journal of Differential Equations, 2012 (2012), 1-22.   Google Scholar [5] R. M. Anderson and R. May, Infectious Diseases of Humans Oxford University Press, New York, 1991. Google Scholar [6] S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example, Int. Stat. Rev., 62 (1994), 229-243.  doi: 10.2307/1403510.  Google Scholar [7] S. B. Cantor and T. G. Ganiats, Incremental cost-effectiveness analysis: The optimal strategy depends on the strategy set, Journal of Clinical Epidemiology, 52 (1999), 517-522.  doi: 10.1016/S0895-4356(99)00021-9.  Google Scholar [8] Centers for Disease Control and Prevention. Chikungunya virus, Available from: http://www.cdc.gov/Chikungunya/symptoms/index.html. Google Scholar [9] L. J. Chang, K. A. Dowd, F. H. Mendoza, J. G. Saunders and S. Sitar, Safety and tolerability of Chikungunya virus-like particle vaccine in healthy adults: A phase 1 dose-escalation trial, Lancet, 384 (2014), 2046-2052.  doi: 10.1016/S0140-6736(14)61185-5.  Google Scholar [10] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar [11] A. Costero, K. Mormann and S. A. Juliano, Asymmetrical competition and patterns of abundance of Aedes albopictus and Culex pipiens (Diptera: Culicidae), Journal of Medical Entomology, 42 (2005), 559-570.   Google Scholar [12] H. Delatte, G. Gimonneau, A. Triboire and D. Fontenille, Influence of temperature on immature development, survival, longevity, fecundity, and gonotrophic cycles of Aedes albopictus, vector of Chikungunya and dengue in the Indian Ocean, Journal of Medical Entomology, 46 (2009), 33-41.   Google Scholar [13] O. Diekmann, J. A. P. Heesterbeek and J. A. P. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar [14] M. Doucleff, Trouble In Paradise: Chikungunya outbreak grows in caribbean, Health News, 2013. Available from: http://www.npr.org/sections/health-shots/2013/12/18/255216192/trouble-in-paradise-Chikungunya-outbreak-grows-in-caribbean. Google Scholar [15] M. Dubrulle, L. Mousson, S. Moutailler, M. Vazeille and A. B. Failloux, Chikungunya virus and Aedes mosquitoes: Saliva is infectious as soon as two days after oral infection, PLoS One, 4 (2009), e5895.  doi: 10.1371/journal.pone.0005895.  Google Scholar [16] Y. Dumont, F. Chiroleu and C. Domerg, On a temporal model for the Chikungunya disease: Modeling, theory and numerics, Mathematical Biosciences, 213 (2008), 80-91.  doi: 10.1016/j.mbs.2008.02.008.  Google Scholar [17] Y. Dumont and F. Chiroleu, Vector control for the Chikungunya disease, Mathematical Biosciences and Engineering, 7 (2010), 313-345.  doi: 10.3934/mbe.2010.7.313.  Google Scholar [18] M. Enserink, Epidemiology: Tropical disease follows mosquitoes to Europe, Science, 317 (2007), 1485a.   Google Scholar [19] K. Fiscella and P. Franks, Cost-effectiveness of the transdermal nicotine patch as an adjunct to physicians smoking cessation counseling, JAMA, 276 (1996), 1247-1251.   Google Scholar [20] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control Springer Verlag, New York, 1975. Google Scholar [21] K. A. Freedberg, J. A. Scharfstein, G. R. Seage, E. Losina, M. C. Weinstein and D. E. Craven, The cost-effectiveness of preventing AIDS-related opportunistic infections, JAMA, 279 (1998), 130-136.  doi: 10.1001/jama.279.2.130.  Google Scholar [22] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar [23] H. R. Joshi, Optimal control of an HIV immunology model, Optim. Control Appl. Math, 23 (2002), 199-213.  doi: 10.1002/oca.710.  Google Scholar [24] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482.  doi: 10.3934/dcdsb.2002.2.473.  Google Scholar [25] D. Kern, S. Lenhart, R. Miller and J. Yong, Optimal control applied to native-invasive population dynamics, J Biol Dyn., 1 (2007), 413-426.  doi: 10.1080/17513750701605556.  Google Scholar [26] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792.  doi: 10.1007/s002850050076.  Google Scholar [27] C. Lahariya and S. K. Pradhan, Emergence of Chikungunya virus in Indian subcontinent after 32 years: A review, Journal of Vector Borne Diseases, 43 (2006), 151-160.   Google Scholar [28] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models Chapman and Hall, 2007. Google Scholar [29] S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar [30] C. Manore, J. Hickmann, S. Xu, H. Wearing and J. Hyman, Comparing Dengue and Chikungunya emergence and endemic Transmission in A. aegypti and A. albopictus, Journal of Theoretical Biology, 356 (2014), 174-191.  doi: 10.1016/j.jtbi.2014.04.033.  Google Scholar [31] E. Massad, S. Ma, M. N. Burattini, Y. Tun, F. A. B. Coutinho and W. Lang, The risk of Chikungunya fever in a dengue-endemic area, Journal of Travel Medicine, 15 (2008), 147-155.  doi: 10.1111/j.1708-8305.2008.00186.x.  Google Scholar [32] R. G. McLeod, J. F. Brewster, A. B. Gumel and D. A. Slonowsky, Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs, Math. Biosci. Eng., 3 (2006), 527-544.  doi: 10.3934/mbe.2006.3.527.  Google Scholar [33] O. P. Misra and D. K. Mishra, Simultaneous effects of control Measures on the transmission dynamics of Chikungunya disease, Applied Mathematics, 2 (2012), 124-130.  doi: 10.5923/j.am.20120204.05.  Google Scholar [34] D. Moulay, M. A. Aziz-Alaoui and M. Cadivel, The Chikungunya disease: Modeling, vector and transmission global dynamics, Mathematical Biosciences, 229 (2011), 50-63.  doi: 10.1016/j.mbs.2010.10.008.  Google Scholar [35] H. Nur Aida, A. Abu Hassan, A. T. Nurita, M. R. Che Salmah and B. Norasmah, Population analysis of Aedes albopictus (Skuse)(Diptera: Culicidae) under uncontrolled laboratory conditions, Tropical Biomedicine, 25 (2008), 117-125.   Google Scholar [36] K. O. Okosun, O. Rachid and N. Marcus, Optimal control strategies and cost-effectiveness analysis of a malaria model, Biosystems, 111 (2013), 83-101.  doi: 10.1016/j.biosystems.2012.09.008.  Google Scholar [37] K. Pesko, C. J. Westbrook, C. N. Mores, L. P. Lounibos and M. H. 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Systematic flow diagram of age-structured Chikungunya Model (1)-(2)
Simulation of the age-structured Chikungunya model (1)-(2) as a function of time when $\mathcal{R}_0 < 1$. (a) Total number of infectious (asymptomatic and symptomatic) juveniles (b) Total number of infectious (asymptomatic and symptomatic) adults (c) Total number of infectious (asymptomatic and symptomatic) seniors (d) Total number of infectious mosquitoes. Parameter values used are as given in Table 2
Simulation of the age-structured Chikungunya model (1)-(2) as a function of time when $\mathcal{R}_0 > 1$. With parameter values used are as given in Table 2. (a) Total number of infectious (asymptomatic and symptomatic) juveniles (b) Total number of infectious (asymptomatic and symptomatic) adults (c) Total number of infectious (asymptomatic and symptomatic) seniors (d) Total number of infectious mosquitoes
PRCC values for the age-structured Chikungunya model (1)-(2), using the basic reproduction number ($\mathcal{R}_{0}$) as the response function. Parameter values (baseline) and ranges used are as given in Table 2
Simulation of the age-structured Chikungunya model (3)-(4) as a function of time without control and with optimal control for: (a). Total number of infected juvenile; (b). Total number of infected adult; (c). Total number of infected seniors; (d). Total number of infected mosquitoes
The optimal controls of the age-structured Chikungunya model (3)-(4) for: (a). Juvenile treatment; (b). Adult treatment; (c). Seniors treatment; (d). Personal protection control; (e). Mosquitoes adulticiding control
Simulation of the age-structured Chikungunya model (3)-(4) as a function of time using strategies A, B and C for: (a). Total number of infected juvenile; (b). Total number of infected adult; (c). Total number of infected seniors; (d). Total number of infected mosquitoes
The optimal controls of the age-structured Chikungunya model (3)-(4) using strategies A, B and C for: (a). Juvenile treatment; (b). Adult treatment; (c). Seniors treatment; (d). Personal protection control; (e). Mosquitoes adulticiding control
Simulation of the age-structured Chikungunya model (3)-(4) as a function of time using strategies D, E and F for: (a). Total number of infected juvenile; (b). Total number of infected adult; (c). Total number of infected seniors; (d). Total number of infected mosquitoes
The optimal controls of the age-structured Chikungunya model (3)-(4) using strategies D, E and F for: (a). Juvenile treatment; (b). Adult treatment; (c). Seniors treatment; (d). Personal protection control; (e). Mosquitoes adulticiding control
Simulation of the age-structured Chikungunya model (3)-(4) as a function of time using strategies B, E and G for: (a). Total number of infected juvenile; (b). Total number of infected adult; (c). Total number of infected seniors; (d). Total number of infected mosquitoes
The optimal controls of the age-structured Chikungunya model (3)-(4) using strategies B, E and G for: (a). Juvenile treatment; (b). Adult treatment; (c). Seniors treatment
Description of the variables and parameters of the agestructured Chikungunya model (1)-(2)
 Variable Description $S_J, S_A, S_S$ Population of susceptible juvenile and adult humans $E_J, E_A, E_S$ Population of exposed juvenile and adult humans $I_{AJ}, I_{SJ}$ Population of asymptomatic and symptomatic Juvenile humans $I_{AA}, I_{SA}$ Population of asymptomatic and symptomatic adult humans $I_{AS}, I_{SS}$ Population of asymptomatic and symptomatic seniors humans $R_J, R_A, R_S$ Population of recovered juvenile and adult humans $S_M$ Population of susceptible mosquitoes $E_M$ Population of exposed mosquitoes $I_{M}$ Population of infected mosquitoes Parameter Description $\pi_J$ Recruitment rate of juvenile humans $\pi_M$ Recruitment rate of mosquitoes $\alpha, \xi$ Juvenile and adult maturation rates $\beta_J, \beta_A, \beta_S$ Transmission probability per contact for susceptible humans $\mu_J, \mu_A, \mu_S$ Natural death rate of juvenile, adult and senior humans $\varepsilon_J, \varepsilon_A, \varepsilon_S$ Fraction of exposed humans becoming asymptomatic $\sigma_J, \sigma_A, \sigma_S$ Progression rate of exposed juvenile, adult and senior humans $\gamma_{AJ},\gamma_{SJ}$ Recovery rate of asymptomatic and symptomatic juvenile humans $\gamma_{AA},\gamma_{SA}$ Recovery rate of asymptomatic and symptomatic adult humans $\gamma_{AS},\gamma_{SS}$ Recovery rate of asymptomatic and symptomatic senior humans $\beta_M$ Transmission probability per contact for susceptible mosquitoes $b_M$ Mosquito biting rate $\sigma_M$ Progression rate of exposed mosquitoes $\mu_M$ Natural death rate of mosquitoes
 Variable Description $S_J, S_A, S_S$ Population of susceptible juvenile and adult humans $E_J, E_A, E_S$ Population of exposed juvenile and adult humans $I_{AJ}, I_{SJ}$ Population of asymptomatic and symptomatic Juvenile humans $I_{AA}, I_{SA}$ Population of asymptomatic and symptomatic adult humans $I_{AS}, I_{SS}$ Population of asymptomatic and symptomatic seniors humans $R_J, R_A, R_S$ Population of recovered juvenile and adult humans $S_M$ Population of susceptible mosquitoes $E_M$ Population of exposed mosquitoes $I_{M}$ Population of infected mosquitoes Parameter Description $\pi_J$ Recruitment rate of juvenile humans $\pi_M$ Recruitment rate of mosquitoes $\alpha, \xi$ Juvenile and adult maturation rates $\beta_J, \beta_A, \beta_S$ Transmission probability per contact for susceptible humans $\mu_J, \mu_A, \mu_S$ Natural death rate of juvenile, adult and senior humans $\varepsilon_J, \varepsilon_A, \varepsilon_S$ Fraction of exposed humans becoming asymptomatic $\sigma_J, \sigma_A, \sigma_S$ Progression rate of exposed juvenile, adult and senior humans $\gamma_{AJ},\gamma_{SJ}$ Recovery rate of asymptomatic and symptomatic juvenile humans $\gamma_{AA},\gamma_{SA}$ Recovery rate of asymptomatic and symptomatic adult humans $\gamma_{AS},\gamma_{SS}$ Recovery rate of asymptomatic and symptomatic senior humans $\beta_M$ Transmission probability per contact for susceptible mosquitoes $b_M$ Mosquito biting rate $\sigma_M$ Progression rate of exposed mosquitoes $\mu_M$ Natural death rate of mosquitoes
Parameters values of the age-structured Chikungunya model (1)-(2)
 Parameter Values Range References $\pi_J$ $\frac{1}{15\times365}$ $\frac{1}{15\times365}$ -$\frac{1}{12\times365}$ [10,30] $\alpha$ $\frac{1}{16\times365}$ $\frac{1}{18\times365}$ -$\frac{1}{15\times365}$ Assumed $\xi$ $\frac{1}{60\times365}$ $\frac{1}{65\times365}$ -$\frac{1}{55\times365}$ Assumed $\beta_J, \beta_A, \beta_S$ 0.24 0.001 -0.54 [16,17,30,40,50] $b_M$ 0.25 0.19 -0.39 [12,30] $\mu_J$ $\frac{1}{3\times365}$ $\frac{1}{5\times365}$ -$\frac{1}{1\times365}$ Assumed $\mu_A$ $\frac{1}{40\times365}$ $\frac{1}{60\times365}$ -$\frac{1}{18\times365}$ Assumed $\mu_S$ $\frac{1}{70\times365}$ $\frac{1}{80\times365}$ -$\frac{1}{60\times365}$ Assumed $\varepsilon_J, \varepsilon_A, \varepsilon_S$ 0.155 0.03-0.28 [48] $\sigma_J, \sigma_S$ $\frac{1}{2\times3}$ $\frac{1}{2\times4}$ -$\frac{1}{2\times2}$ Assumed $\sigma_A$ $\frac{1}{3}$ $\frac{1}{4}$ -$\frac{1}{2}$ [17,27,30,38,46,45] $\gamma_{AJ},\gamma_{SJ}$ $\frac{1}{1.5\times4.5}$ $\frac{1}{1.5\times8}$ -$\frac{1}{1.5\times3}$ Assumed $\gamma_{AA},\gamma_{SA}$ $\frac{1}{4.5}$ $\frac{1}{7}$ -$\frac{1}{3}$ [30,34,46,45] $\gamma_{AS},\gamma_{SS}$ $\frac{1}{2.5\times4.5}$ $\frac{1}{2.5\times8}$ -$\frac{1}{2.5\times3}$ Assumed $\pi_M$ 0.24 0.015 -0.32 [10,11,30,35] $\beta_M$ 0.24 0.005 -0.35 [16,31,37,40,50] $\sigma_M$ $\frac{1}{3.5}$ $\frac{1}{6}$ -$\frac{1}{2}$ [15,17,34,46] $\mu_M$ $\frac{1}{14}$ $\frac{1}{42}$ -$\frac{1}{14}$ [17,27,34,46,45]
 Parameter Values Range References $\pi_J$ $\frac{1}{15\times365}$ $\frac{1}{15\times365}$ -$\frac{1}{12\times365}$ [10,30] $\alpha$ $\frac{1}{16\times365}$ $\frac{1}{18\times365}$ -$\frac{1}{15\times365}$ Assumed $\xi$ $\frac{1}{60\times365}$ $\frac{1}{65\times365}$ -$\frac{1}{55\times365}$ Assumed $\beta_J, \beta_A, \beta_S$ 0.24 0.001 -0.54 [16,17,30,40,50] $b_M$ 0.25 0.19 -0.39 [12,30] $\mu_J$ $\frac{1}{3\times365}$ $\frac{1}{5\times365}$ -$\frac{1}{1\times365}$ Assumed $\mu_A$ $\frac{1}{40\times365}$ $\frac{1}{60\times365}$ -$\frac{1}{18\times365}$ Assumed $\mu_S$ $\frac{1}{70\times365}$ $\frac{1}{80\times365}$ -$\frac{1}{60\times365}$ Assumed $\varepsilon_J, \varepsilon_A, \varepsilon_S$ 0.155 0.03-0.28 [48] $\sigma_J, \sigma_S$ $\frac{1}{2\times3}$ $\frac{1}{2\times4}$ -$\frac{1}{2\times2}$ Assumed $\sigma_A$ $\frac{1}{3}$ $\frac{1}{4}$ -$\frac{1}{2}$ [17,27,30,38,46,45] $\gamma_{AJ},\gamma_{SJ}$ $\frac{1}{1.5\times4.5}$ $\frac{1}{1.5\times8}$ -$\frac{1}{1.5\times3}$ Assumed $\gamma_{AA},\gamma_{SA}$ $\frac{1}{4.5}$ $\frac{1}{7}$ -$\frac{1}{3}$ [30,34,46,45] $\gamma_{AS},\gamma_{SS}$ $\frac{1}{2.5\times4.5}$ $\frac{1}{2.5\times8}$ -$\frac{1}{2.5\times3}$ Assumed $\pi_M$ 0.24 0.015 -0.32 [10,11,30,35] $\beta_M$ 0.24 0.005 -0.35 [16,31,37,40,50] $\sigma_M$ $\frac{1}{3.5}$ $\frac{1}{6}$ -$\frac{1}{2}$ [15,17,34,46] $\mu_M$ $\frac{1}{14}$ $\frac{1}{42}$ -$\frac{1}{14}$ [17,27,34,46,45]
Incremental cost-effectiveness ratio in increasing order of total infection averted
 Strategies Total infection averted Total Cost ICER Strategy C $2.4390\times 10^{6}$ $9.3014\times 10^{7}$ $38.1361$ Strategy A $2.7536\times 10^{6}$ $6.5306\times 10^{7}$ $-88.0737$ Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $-76.1028$
 Strategies Total infection averted Total Cost ICER Strategy C $2.4390\times 10^{6}$ $9.3014\times 10^{7}$ $38.1361$ Strategy A $2.7536\times 10^{6}$ $6.5306\times 10^{7}$ $-88.0737$ Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $-76.1028$
Incremental cost-effectiveness ratio in increasing order of total infection averted
 Strategies Total infection averted Total Cost ICER Strategy A $2.7536\times 10^{6}$ $6.5306\times 10^{7}$ $23.7166$ Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $-76.1028$
 Strategies Total infection averted Total Cost ICER Strategy A $2.7536\times 10^{6}$ $6.5306\times 10^{7}$ $23.7166$ Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $-76.1028$
Incremental cost-effectiveness ratio in increasing order of total infection averted
 Strategies Total infection averted Total Cost ICER Strategy F $2.7536\times 10^{6}$ $6.1476\times 10^{7}$ $22.32536$ Strategy E $3.3444\times 10^{6}$ $3.3421\times 10^{6}$ $-98.3986$ Strategy D $3.3475\times 10^{6}$ $6.5525\times 10^{6}$ $1035.6129$
 Strategies Total infection averted Total Cost ICER Strategy F $2.7536\times 10^{6}$ $6.1476\times 10^{7}$ $22.32536$ Strategy E $3.3444\times 10^{6}$ $3.3421\times 10^{6}$ $-98.3986$ Strategy D $3.3475\times 10^{6}$ $6.5525\times 10^{6}$ $1035.6129$
Incremental cost-effectiveness ratio in increasing order of total infection averted
 Strategies Total infection averted Total Cost ICER Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $6.7470$ Strategy E $3.3444\times 10^{6}$ $3.3421\times 10^{6}$ $-710.5187$ Strategy G $3.3475\times 10^{6}$ $2.7131\times 10^{6}$ $-202.9032$
 Strategies Total infection averted Total Cost ICER Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $6.7470$ Strategy E $3.3444\times 10^{6}$ $3.3421\times 10^{6}$ $-710.5187$ Strategy G $3.3475\times 10^{6}$ $2.7131\times 10^{6}$ $-202.9032$
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