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May  2017, 22(3): 977-1000. doi: 10.3934/dcdsb.2017049

## Management strategies in a malaria model combining human and transmission-blocking vaccines

 1 Department of Mathematics, Valdosta State University, Valdosta, GA 31698, USA 2 Department of Mathematics and Statistics, Minnesota State University, Mankato, Mankato, MN, 56001, USA 3 Department of Mathematics, Augusta University, Augusta, GA 30912, USA 4 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

Received  August 2015 Revised  September 2016 Published  January 2017

We propose a new mathematical model studying control strategies of malaria transmission. The control is a combination of human and transmission-blocking vaccines and vector control (larvacide). When the disease induced death rate is large enough, we show the existence of a backward bifurcation analytically if vaccination control is not used, and numerically if vaccination is used. The basic reproduction number is a decreasing function of the vaccination controls as well as the vector control parameters, which means that any effort on these controls will reduce the burden of the disease. Numerical simulation suggests that the combination of the vaccinations and vector control may help to eradicate the disease. We investigate optimal strategies using the vaccinations and vector controls to gain qualitative understanding on how the combinations of these controls should be used to reduce disease prevalence in malaria endemic setting. Our results show that the combination of the two vaccination controls integrated with vector control has the highest impact on reducing the number of infected humans and mosquitoes.

Citation: Jemal Mohammed-Awel, Ruijun Zhao, Eric Numfor, Suzanne Lenhart. Management strategies in a malaria model combining human and transmission-blocking vaccines. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 977-1000. doi: 10.3934/dcdsb.2017049
##### References:
  S. Ai, J. Li and J. Lu, Mosquito-stage-structured malaria models and their global dynamics, SIAM J. Appl. Math., 72 (2012), 1213-1237.  doi: 10.1137/110860318.  Google Scholar  M. Arevalo-Herrera, Y. Solarte, C. Marin, M. Santos, J. Castellanos, J. C. Beier and S. H. Valencia, Malaria transmission blocking immunity and sexual stage vaccines for interrupting malaria transmission in Latin America, Mem Inst Oswaldo Cruz, Rio de Janeiro., 106 (2011), 202-211.  doi: 10.1590/S0074-02762011000900025. Google Scholar  J. Arino, A. Ducrot and P. Zongo, A metapopulation model for malaria with transmission-blocking partial immunity in hosts, J. Math. Biol., 64 (2012), 423-448.  doi: 10.1007/s00285-011-0418-4.  Google Scholar  A. J. Birkett, V. S. Moorthy, C. Loucq, C. E. Chitnis and D. C. Kaslow, Malaria vaccine R&D in the Decade of Vaccines: Breakthroughs, challenges and opportunities, Vaccine, 31 (2013), B233-B243.  doi: 10.1016/j.vaccine.2013.02.040. Google Scholar  R. Carter, Transmission blocking malaria vaccines, Vaccine, 19 (2001), 2309-2314.  doi: 10.1016/S0264-410X(00)00521-1. Google Scholar  M. C. de Castro, Y. Yamagata, D. Mtasiwa, M. Tanner, J. Utzinger, J. Keiser and B. H. Singer, Integrated urban malaria control: A case study in Dar es Salaam, Tanzania, Am. J. Trop. Med. Hyg., 71 (2004), 103-117.   Google Scholar  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, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar  O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R o in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar  P. V. den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar  X. Feng, S. Ruan, Z. Teng and K. Wang, Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Biosci., 266 (2015), 52-64.  doi: 10.1016/j.mbs.2015.05.005.  Google Scholar  K. R Fister, S. Lenhart and J. S McNally, Optimizing Chemotherapy in an HIV Model, Electron. J. Diff. Eqns., (1998), 1-12. Google Scholar  S. Gandon, M. J. Mackinnon, S. Nee and A. F. Read, Imperfect vaccines and the evolution of pathogen virulence, Nature, 414 (2001), 751-756.   Google Scholar  A. B. Gumel, Causes of backward bifurcations in some epidemiological models, J. Math. Anal. Appl., 395 (2012), 355-365.  doi: 10.1016/j.jmaa.2012.04.077.  Google Scholar  M. E. Halloran and C. J. Struchiner, Modeling transmission dynamics of stage-specific malaria vaccines, Parasitology Today, 8 (1992), 77-85.  doi: 10.1016/0169-4758(92)90240-3. Google Scholar  E coli has applications for malaria vaccine, HematologyTimes 2014. Available from: http://www.hematologytimes.com/p_article.do?id=3845 Google Scholar  S. Lenhart and J. T Workman, Optimal Control Applied to Biological Models Chapman and Hall, 2007. Google Scholar  N. C. Ngonghalaa, G. A. Ngwa and M. I. Teboh-Ewungkem, Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission, Math. Biosci., 240 (2012), 45-62.  doi: 10.1016/j.mbs.2012.06.003.  Google Scholar  V. Nussenzweig, M. F. Good and A. V. Hill, Mixed results for a malaria vaccine, Nature Medicine, 17 (2011), 1560-1561.   Google Scholar  L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes Wiley, New York, 2002. Google Scholar  O. Prosper, N. Ruktanonchai and M. Martcheva, Optimal vaccination and bednet maintenance for the control of malaria in a region with naturally acquired immunity, J Theor. Biol., 353 (2014), 142-156.  doi: 10.1016/j.jtbi.2014.03.013.  Google Scholar  K. Raghavendra, T. K. Barik, B. P. N. Reddy, P. Sharma and A. P. Dash, Malaria vector control: From past to future, Parasitology Research, 108 (2011), 757-779.  doi: 10.1007/s00436-010-2232-0. Google Scholar  First Results of Phase 3 Trial of RTS, S/AS01 Malaria Vaccine in African Children, The RTS, S Clinical Trials Partnership, N. Engl. J. Med. , 365 (2011), 1863–1875. Google Scholar  A Phase 3 Trial of RTS, S/AS01 Malaria Vaccine in African Infants, The RTS, S Clinical Trials Partnership, N. Engl. J. Med. , 367 (2012), 2284–2295. Google Scholar  A. Saul, Mosquito stage, transmission blocking vaccines for malaria, Curr. Opin. Infect. Dis, 20 (2007), 476-481.  doi: 10.1097/QCO.0b013e3282a95e12. Google Scholar  M. K. Seo, P. Baker and K. N. Ngo, Cost-effectiveness analysis of vaccinating children in Malawi with RTS, S vaccines in comparison with long-lasting insecticide-treated nets, Malaria Journal, 13 (2014), 66-76.  doi: 10.1186/1475-2875-13-66. Google Scholar  B. Sharma, Structure and mechanism of a transmission blocking vaccine candidate protein Pfs25 from P. falciparum: a molecular modeling and docking study, In Silico Biol., 8 (2008), 193-206.   Google Scholar  R. J. Smith, Mathematical models of malaria -a review.Could Low-Efficacy Malaria Vaccines Increase Secondary Infections in Endemic Areas. Mathematical Modeling of Biological Systems Volume Ⅱ Modeling and Simulation in Science, Engineering and Technology, Mathematical Modeling of Biological Systems, Volume Ⅱ, 2 (2008), 3-9.   Google Scholar  T. A. Smith, N. Chitnis and M. Tanner, Uses of mosquito-stage transmission-blocking vaccines against plasmodium falciparum, Trends in Parasitology, 27 (2011), 190-196.  doi: 10.1016/j.pt.2010.12.011. Google Scholar  T. Smith, G. F. Killeen, N. Maire, A. Ross, L. Molineaux, F. Tediosi, G. Hutton, J. Utzinger, K. Dietz and A. M. Tanner, Mathematical Modeling of The Impact of Malaria Vaccines on The Clinical Epidemiology and Natural History of Plasmodium Falciparum Malaria: Overview, Am. J. Trop. Med. Hyg., 75 (2006), 1-10.   Google Scholar  M. T Teboh-Ewungkem, C. N. Podder and A. Gumel, Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics, Bull. Math. Biol., 72 (2010), 63-93.  doi: 10.1007/s11538-009-9437-3.  Google Scholar  N. J. White, A vaccine for malaria, N. Engl. J. Med., 365 (2011), 1926-1927.  doi: 10.1056/NEJMe1111777. Google Scholar  World Health Organization 2000: Malaria transmission blocking vaccines: An ideal public good (2000) Available from: http://www.who.int/tdr/publications/tdr-research-publications/malaria-transmission-blocking-vaccines/en/. Google Scholar  World Malaria Report (2015) Available from: http://www.who.int/malaria/media/world-malaria-report-2015/en/. Google Scholar  R. Zhao and J. Mohammed-Awel, A mathematical model studying mosquito-stage transmission-blocking vaccines, Math. Biosci. Eng., 11 (2014), 1229-1245.  doi: 10.3934/mbe.2014.11.1229.  Google Scholar

show all references

##### References:
  S. Ai, J. Li and J. Lu, Mosquito-stage-structured malaria models and their global dynamics, SIAM J. Appl. Math., 72 (2012), 1213-1237.  doi: 10.1137/110860318.  Google Scholar  M. Arevalo-Herrera, Y. Solarte, C. Marin, M. Santos, J. Castellanos, J. C. Beier and S. H. Valencia, Malaria transmission blocking immunity and sexual stage vaccines for interrupting malaria transmission in Latin America, Mem Inst Oswaldo Cruz, Rio de Janeiro., 106 (2011), 202-211.  doi: 10.1590/S0074-02762011000900025. Google Scholar  J. Arino, A. Ducrot and P. Zongo, A metapopulation model for malaria with transmission-blocking partial immunity in hosts, J. Math. Biol., 64 (2012), 423-448.  doi: 10.1007/s00285-011-0418-4.  Google Scholar  A. J. Birkett, V. S. Moorthy, C. Loucq, C. E. Chitnis and D. C. Kaslow, Malaria vaccine R&D in the Decade of Vaccines: Breakthroughs, challenges and opportunities, Vaccine, 31 (2013), B233-B243.  doi: 10.1016/j.vaccine.2013.02.040. Google Scholar  R. Carter, Transmission blocking malaria vaccines, Vaccine, 19 (2001), 2309-2314.  doi: 10.1016/S0264-410X(00)00521-1. Google Scholar  M. C. de Castro, Y. Yamagata, D. Mtasiwa, M. Tanner, J. Utzinger, J. Keiser and B. H. Singer, Integrated urban malaria control: A case study in Dar es Salaam, Tanzania, Am. J. Trop. Med. Hyg., 71 (2004), 103-117.   Google Scholar  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, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar  O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R o in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar  P. V. den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar  X. Feng, S. Ruan, Z. Teng and K. Wang, Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Biosci., 266 (2015), 52-64.  doi: 10.1016/j.mbs.2015.05.005.  Google Scholar  K. R Fister, S. Lenhart and J. S McNally, Optimizing Chemotherapy in an HIV Model, Electron. J. Diff. Eqns., (1998), 1-12. Google Scholar  S. Gandon, M. J. Mackinnon, S. Nee and A. F. Read, Imperfect vaccines and the evolution of pathogen virulence, Nature, 414 (2001), 751-756.   Google Scholar  A. B. Gumel, Causes of backward bifurcations in some epidemiological models, J. Math. Anal. Appl., 395 (2012), 355-365.  doi: 10.1016/j.jmaa.2012.04.077.  Google Scholar  M. E. Halloran and C. J. Struchiner, Modeling transmission dynamics of stage-specific malaria vaccines, Parasitology Today, 8 (1992), 77-85.  doi: 10.1016/0169-4758(92)90240-3. Google Scholar  E coli has applications for malaria vaccine, HematologyTimes 2014. Available from: http://www.hematologytimes.com/p_article.do?id=3845 Google Scholar  S. Lenhart and J. T Workman, Optimal Control Applied to Biological Models Chapman and Hall, 2007. Google Scholar  N. C. Ngonghalaa, G. A. Ngwa and M. I. Teboh-Ewungkem, Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission, Math. Biosci., 240 (2012), 45-62.  doi: 10.1016/j.mbs.2012.06.003.  Google Scholar  V. Nussenzweig, M. F. Good and A. V. Hill, Mixed results for a malaria vaccine, Nature Medicine, 17 (2011), 1560-1561.   Google Scholar  L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes Wiley, New York, 2002. Google Scholar  O. Prosper, N. Ruktanonchai and M. Martcheva, Optimal vaccination and bednet maintenance for the control of malaria in a region with naturally acquired immunity, J Theor. Biol., 353 (2014), 142-156.  doi: 10.1016/j.jtbi.2014.03.013.  Google Scholar  K. Raghavendra, T. K. Barik, B. P. N. Reddy, P. Sharma and A. P. Dash, Malaria vector control: From past to future, Parasitology Research, 108 (2011), 757-779.  doi: 10.1007/s00436-010-2232-0. Google Scholar  First Results of Phase 3 Trial of RTS, S/AS01 Malaria Vaccine in African Children, The RTS, S Clinical Trials Partnership, N. Engl. J. Med. , 365 (2011), 1863–1875. Google Scholar  A Phase 3 Trial of RTS, S/AS01 Malaria Vaccine in African Infants, The RTS, S Clinical Trials Partnership, N. Engl. J. Med. , 367 (2012), 2284–2295. Google Scholar  A. Saul, Mosquito stage, transmission blocking vaccines for malaria, Curr. Opin. Infect. Dis, 20 (2007), 476-481.  doi: 10.1097/QCO.0b013e3282a95e12. Google Scholar  M. K. Seo, P. Baker and K. N. Ngo, Cost-effectiveness analysis of vaccinating children in Malawi with RTS, S vaccines in comparison with long-lasting insecticide-treated nets, Malaria Journal, 13 (2014), 66-76.  doi: 10.1186/1475-2875-13-66. Google Scholar  B. Sharma, Structure and mechanism of a transmission blocking vaccine candidate protein Pfs25 from P. falciparum: a molecular modeling and docking study, In Silico Biol., 8 (2008), 193-206.   Google Scholar  R. J. Smith, Mathematical models of malaria -a review.Could Low-Efficacy Malaria Vaccines Increase Secondary Infections in Endemic Areas. Mathematical Modeling of Biological Systems Volume Ⅱ Modeling and Simulation in Science, Engineering and Technology, Mathematical Modeling of Biological Systems, Volume Ⅱ, 2 (2008), 3-9.   Google Scholar  T. A. Smith, N. Chitnis and M. Tanner, Uses of mosquito-stage transmission-blocking vaccines against plasmodium falciparum, Trends in Parasitology, 27 (2011), 190-196.  doi: 10.1016/j.pt.2010.12.011. Google Scholar  T. Smith, G. F. Killeen, N. Maire, A. Ross, L. Molineaux, F. Tediosi, G. Hutton, J. Utzinger, K. Dietz and A. M. Tanner, Mathematical Modeling of The Impact of Malaria Vaccines on The Clinical Epidemiology and Natural History of Plasmodium Falciparum Malaria: Overview, Am. J. Trop. Med. Hyg., 75 (2006), 1-10.   Google Scholar  M. T Teboh-Ewungkem, C. N. Podder and A. Gumel, Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics, Bull. Math. Biol., 72 (2010), 63-93.  doi: 10.1007/s11538-009-9437-3.  Google Scholar  N. J. White, A vaccine for malaria, N. Engl. J. Med., 365 (2011), 1926-1927.  doi: 10.1056/NEJMe1111777. Google Scholar  World Health Organization 2000: Malaria transmission blocking vaccines: An ideal public good (2000) Available from: http://www.who.int/tdr/publications/tdr-research-publications/malaria-transmission-blocking-vaccines/en/. Google Scholar  World Malaria Report (2015) Available from: http://www.who.int/malaria/media/world-malaria-report-2015/en/. Google Scholar  R. Zhao and J. Mohammed-Awel, A mathematical model studying mosquito-stage transmission-blocking vaccines, Math. Biosci. Eng., 11 (2014), 1229-1245.  doi: 10.3934/mbe.2014.11.1229.  Google Scholar The bifurcation diagram for the special case where $\xi_h = 0$. The basic reproduction number is calculated with $u$ varying. The two panels in the first row show a backward bifurcation for $\delta_h = 4\mu_h$, which might represent severe malaria version. The two panels in the second row show a forward bifurcation for $\delta_h = \mu_h$, which might represent a mild version of malaria. The other parameters are listed in Table 1. On the graphs, solid red lines represent the stable endemic equilibrium, dashed blue lines represent the unstable endemic equilibrium, solid green lines represent the stable disease-free equilibrium and dashed green lines represent the unstable disease-free equilibrium The bifurcation diagram for the complete model. The basic reproduction number is calculated with $u = 0$ and $\xi_h$ varying from $0$ to $0.2$. The three panels in the first row show a backward bifurcation when $\delta_h = 4\mu_h$, which might represent severe malaria version. The three panels in the second row show a forward bifurcation when $\delta_h = 0.5\mu_h$, which might represent a mild version of malaria. On the graphs, solid red lines represent the stable endemic equilibrium, dashed blue lines represent the unstable endemic equilibrium, solid green lines represent the stable disease-free equilibrium and dashed green lines represent the unstable disease-free equilibrium Simulation for different initial data when $\delta_h = 4\mu_h$, $u = 0$, and $\xi_h = 0.03$. The basic reproduction number is $\mathcal{R}_{vac} = 0.95907$. The three panels in the first row show the disease persists for some initial data. The three panels in the second row show the disease asymptotically dies out for some other initial data Simulation for different vaccination effort when $\delta_h = 4\mu_h$ and $u = 0$. The three panels in the first row are generated when $\xi_h = 0.001$, in which $\mathcal{R}_{vac} = 1.62127$, and the disease persists. The three panels in the second row are generated when $\xi_h = 0.1$, in which $\mathcal{R}_{vac} = 0.45893$, and the disease asymptotically dies out Optimal control solution for Case 5 in which the initial values are set to be the endemic equilibrium when no control is applied Optimal control solution for Case 5 control strategy with low level of infections as initial values for the state variables: $S_h^0=4000$, $I_h^0=10$, $R_h^0=2$, $V_B^0=0$, $J_h^0=0$, $M_h^0=0$, $R_h^0=0$, $S_v^0=10000$, $I_v^0=10$, and $V_v^0=0$
Description of parameters of the basic malaria model, the detailed reference for each value can be found in [7,34]
 Parameter Description Baseline values and range $\Lambda_{h}$ Recruitment rate of $10^4/55 \in[10^{4}/72, 10^{4}/35]$ $L$ environmental carrying capacity per year $10^5 \in[10^{4}/72, 10^{4}/35]\times\frac{40}{3}$ $r_v$ mosquito growth constant per year $4\times365/21$ $\mu_{h}$ Natural death rate of host per year $1/55\in[1/72, 1/35]$ $\mu_{v}$ Natural death rate of vector per year $365/21\in[365/28, 365/14]$ $C_{hv}$ The effective transmission rate from humans to mosquitoes per year per bite $9\in[2.6, 32\times365]$ $C_{vh}$ The effective transmission rate from mosquitoes to humans per year per bite $0.8\in[0.001\times365, 0.27\times365]$ $\xi_h$ Vaccination rate of humans with mixture dose per year $0.01\in[0, \ln 5]$ $\rho_{h}$ Rate of loss of immunity per year $2\in[1/50, 4]$ $\eta_{h}$ Rate of development of temporal immunity per year $1\in [1/2, 6]$ $\delta_h$ Disease-induced death rate per year $0.1/55\in[0, 4.1\times10^{-4}]\times365$ $\nu_{h}$ Vaccination rate of humans per year with HV dose only $0.1 \in[0, \ln 5]$ $\omega_{h}$ Rate of loss of HV acquired-immunity per year in vaccinated group of humans $1/4\in[1/5, 1]$ $u$ $1-u$ represents mosquitoes birth due to control such asreduction factor of larvacide $0\in[0, 0.72]$
 Parameter Description Baseline values and range $\Lambda_{h}$ Recruitment rate of $10^4/55 \in[10^{4}/72, 10^{4}/35]$ $L$ environmental carrying capacity per year $10^5 \in[10^{4}/72, 10^{4}/35]\times\frac{40}{3}$ $r_v$ mosquito growth constant per year $4\times365/21$ $\mu_{h}$ Natural death rate of host per year $1/55\in[1/72, 1/35]$ $\mu_{v}$ Natural death rate of vector per year $365/21\in[365/28, 365/14]$ $C_{hv}$ The effective transmission rate from humans to mosquitoes per year per bite $9\in[2.6, 32\times365]$ $C_{vh}$ The effective transmission rate from mosquitoes to humans per year per bite $0.8\in[0.001\times365, 0.27\times365]$ $\xi_h$ Vaccination rate of humans with mixture dose per year $0.01\in[0, \ln 5]$ $\rho_{h}$ Rate of loss of immunity per year $2\in[1/50, 4]$ $\eta_{h}$ Rate of development of temporal immunity per year $1\in [1/2, 6]$ $\delta_h$ Disease-induced death rate per year $0.1/55\in[0, 4.1\times10^{-4}]\times365$ $\nu_{h}$ Vaccination rate of humans per year with HV dose only $0.1 \in[0, \ln 5]$ $\omega_{h}$ Rate of loss of HV acquired-immunity per year in vaccinated group of humans $1/4\in[1/5, 1]$ $u$ $1-u$ represents mosquitoes birth due to control such asreduction factor of larvacide $0\in[0, 0.72]$
Values for the parameters for our numerical scenarios
 Parameter Value Parameter Value Parameter Value Parameter Value A1 7.3 B1 A1 C1 100 $\xi _h^{\max }$ ln(4)/4 A2 A1 B2 B1/2 C2 100 $\nu _h^{\max }$ ln(4)/4 A3 A1/10 B3 B1/100 C3 100 ${u^{\max }}$ 0.4
 Parameter Value Parameter Value Parameter Value Parameter Value A1 7.3 B1 A1 C1 100 $\xi _h^{\max }$ ln(4)/4 A2 A1 B2 B1/2 C2 100 $\nu _h^{\max }$ ln(4)/4 A3 A1/10 B3 B1/100 C3 100 ${u^{\max }}$ 0.4
Values of the objective functional at the optimal control solution (column two) and at the upper bound values of the nonzero controls (column three); the fourth column is the percentage decrease in cost for each strategy at optimal control compared to strategy with maximum control
 Strategy/Control Cost with optimal control Cost with constant maximum control Percentage decrease in cost at Optimal Control compared to maximum control Case 1 181, 345 191, 066 5.09% Case 2 199, 938 199, 943 0.0025% Case 3 157, 672 171, 830 8.24% Case 4 181, 129 193, 335 6.31% Case 5 157, 635 175, 596 10.23%
 Strategy/Control Cost with optimal control Cost with constant maximum control Percentage decrease in cost at Optimal Control compared to maximum control Case 1 181, 345 191, 066 5.09% Case 2 199, 938 199, 943 0.0025% Case 3 157, 672 171, 830 8.24% Case 4 181, 129 193, 335 6.31% Case 5 157, 635 175, 596 10.23%
Values of the cost functional at the optimal control solution(column two); percentage cost decrease of each strategy at optimal control compared to strategy without control, Case 0 (column three); and percentage decrease in cost for each strategy at optimal control compared to strategy in Case 2 at optimal control, (column four).
 Strategy/Control Cost with optimal control Percentage cost decrease compared with Case 0 Percentage decrease in cost compared with Case 2 Case 1 181, 345 27.26% 10.25% Case 2 199, 938 15.43% 0.0% Case 3 157, 672 46.37% 26.81% Case 4 181, 129 27.42% 10.38% Case 5 157, 635 46.41% 26.84%
 Strategy/Control Cost with optimal control Percentage cost decrease compared with Case 0 Percentage decrease in cost compared with Case 2 Case 1 181, 345 27.26% 10.25% Case 2 199, 938 15.43% 0.0% Case 3 157, 672 46.37% 26.81% Case 4 181, 129 27.42% 10.38% Case 5 157, 635 46.41% 26.84%
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