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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:
[1]

S. AiJ. 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.

[2]

M. Arevalo-HerreraY. SolarteC. MarinM. SantosJ. CastellanosJ. 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.

[3]

J. ArinoA. 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.

[4]

A. J. BirkettV. S. MoorthyC. LoucqC. 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.

[5]

R. Carter, Transmission blocking malaria vaccines, Vaccine, 19 (2001), 2309-2314. doi: 10.1016/S0264-410X(00)00521-1.

[6]

M. C. de CastroY. YamagataD. MtasiwaM. TannerJ. UtzingerJ. 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.

[7]

N. ChitnisJ. 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.

[8]

O. DiekmannJ. 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.

[9]

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.

[10]

X. FengS. RuanZ. 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.

[11]

K. R FisterS. Lenhart and J. S McNally, Optimizing Chemotherapy in an HIV Model, Electron. J. Diff. Eqns., (1998), 1-12.

[12]

S. GandonM. J. MackinnonS. Nee and A. F. Read, Imperfect vaccines and the evolution of pathogen virulence, Nature, 414 (2001), 751-756.

[13]

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.

[14]

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.

[15]

E coli has applications for malaria vaccine, HematologyTimes 2014. Available from: http://www.hematologytimes.com/p_article.do?id=3845

[16]

S. Lenhart and J. T Workman, Optimal Control Applied to Biological Models Chapman and Hall, 2007.

[17]

N. C. NgonghalaaG. 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.

[18]

V. NussenzweigM. F. Good and A. V. Hill, Mixed results for a malaria vaccine, Nature Medicine, 17 (2011), 1560-1561.

[19]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes Wiley, New York, 2002.

[20]

O. ProsperN. 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.

[21]

K. RaghavendraT. K. BarikB. P. N. ReddyP. 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.

[22]

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.

[23]

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.

[24]

A. Saul, Mosquito stage, transmission blocking vaccines for malaria, Curr. Opin. Infect. Dis, 20 (2007), 476-481. doi: 10.1097/QCO.0b013e3282a95e12.

[25]

M. K. SeoP. 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.

[26]

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.

[27]

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.

[28]

T. A. SmithN. 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.

[29]

T. SmithG. F. KilleenN. MaireA. RossL. MolineauxF. TediosiG. HuttonJ. UtzingerK. 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.

[30]

M. T Teboh-EwungkemC. 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.

[31]

N. J. White, A vaccine for malaria, N. Engl. J. Med., 365 (2011), 1926-1927. doi: 10.1056/NEJMe1111777.

[32]

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/.

[33]

World Malaria Report (2015) Available from: http://www.who.int/malaria/media/world-malaria-report-2015/en/.

[34]

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.

show all references

References:
[1]

S. AiJ. 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.

[2]

M. Arevalo-HerreraY. SolarteC. MarinM. SantosJ. CastellanosJ. 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.

[3]

J. ArinoA. 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.

[4]

A. J. BirkettV. S. MoorthyC. LoucqC. 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.

[5]

R. Carter, Transmission blocking malaria vaccines, Vaccine, 19 (2001), 2309-2314. doi: 10.1016/S0264-410X(00)00521-1.

[6]

M. C. de CastroY. YamagataD. MtasiwaM. TannerJ. UtzingerJ. 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.

[7]

N. ChitnisJ. 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.

[8]

O. DiekmannJ. 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.

[9]

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.

[10]

X. FengS. RuanZ. 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.

[11]

K. R FisterS. Lenhart and J. S McNally, Optimizing Chemotherapy in an HIV Model, Electron. J. Diff. Eqns., (1998), 1-12.

[12]

S. GandonM. J. MackinnonS. Nee and A. F. Read, Imperfect vaccines and the evolution of pathogen virulence, Nature, 414 (2001), 751-756.

[13]

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.

[14]

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.

[15]

E coli has applications for malaria vaccine, HematologyTimes 2014. Available from: http://www.hematologytimes.com/p_article.do?id=3845

[16]

S. Lenhart and J. T Workman, Optimal Control Applied to Biological Models Chapman and Hall, 2007.

[17]

N. C. NgonghalaaG. 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.

[18]

V. NussenzweigM. F. Good and A. V. Hill, Mixed results for a malaria vaccine, Nature Medicine, 17 (2011), 1560-1561.

[19]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes Wiley, New York, 2002.

[20]

O. ProsperN. 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.

[21]

K. RaghavendraT. K. BarikB. P. N. ReddyP. 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.

[22]

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.

[23]

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.

[24]

A. Saul, Mosquito stage, transmission blocking vaccines for malaria, Curr. Opin. Infect. Dis, 20 (2007), 476-481. doi: 10.1097/QCO.0b013e3282a95e12.

[25]

M. K. SeoP. 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.

[26]

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.

[27]

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.

[28]

T. A. SmithN. 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.

[29]

T. SmithG. F. KilleenN. MaireA. RossL. MolineauxF. TediosiG. HuttonJ. UtzingerK. 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.

[30]

M. T Teboh-EwungkemC. 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.

[31]

N. J. White, A vaccine for malaria, N. Engl. J. Med., 365 (2011), 1926-1927. doi: 10.1056/NEJMe1111777.

[32]

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/.

[33]

World Malaria Report (2015) Available from: http://www.who.int/malaria/media/world-malaria-report-2015/en/.

[34]

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.

Figure 1.  The schematic diagram of the mathematical model
Figure 2.  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
Figure 3.  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
Figure 4.  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
Figure 5.  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
Figure 6.  Optimal control solution for Case 5 in which the initial values are set to be the endemic equilibrium when no control is applied
Figure 7.  Optimal solution of human and mosquito population for Case 5
Figure 8.  Optimal control solution for Case 4
Figure 9.  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 $V_v^0=0$
Table 1.  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 $0\in[0, 0.72]$
Table 2.  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
Table 3.  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%
Table 4.  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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