# American Institute of Mathematical Sciences

December  2011, 1(4): 493-508. doi: 10.3934/mcrf.2011.1.493

## Optimal control of a vector-host epidemics model

 1 School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China, China, China 2 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, China

Received  June 2011 Revised  September 2011 Published  November 2011

In this paper, a vector-host epidemic model with control measures is considered to assess the impact of control measures on the prevalence of the vector-host diseases. We incorporated mosquito-reduction strategy and host medical treatment into the model. For the basic vector-host model, we provide sufficient conditions for the local stability of the disease free equilibrium (DFE) and the sensitivity analysis for the reproduction number with respect to the model parameters. Using the optimal control theory, the optimal levels of the two controls are characterized, and then the existence and uniqueness for the optimal control pair are established. Numerical simulations are further conducted to confirm and extend the analytical results. Numerical results suggest that optimal multi-control strategy is a more beneficial choice in fighting the outbreak of vector-host diseases. For the vector-host epidemics, vector control measures should be taken prior to other measures.
Citation: Qingkai Kong, Zhipeng Qiu, Zi Sang, Yun Zou. Optimal control of a vector-host epidemics model. Mathematical Control & Related Fields, 2011, 1 (4) : 493-508. doi: 10.3934/mcrf.2011.1.493
##### References:
 [1] World Health Organization Expert Committee on Malaria, 20th Report,, WHO Regional Office for Africa, (2003). Google Scholar [2] S. W. Lindsay, W. J. M. Martens, Malaria in the African highlands: Past, present and future,, Bull. WHO, 76 (1998), 33. Google Scholar [3] G. Zhou, N. Minakawa, A. K. Githeko and G. Yan, Association between climate variability and malaria epidemics in the east African highlands,, Proc. Natl. Acad. Sci., 101 (2004), 2375. doi: 10.1073/pnas.0308714100. Google Scholar [4] Herbert W. Hethcote, The mathematics of infectious diseases,, Socity for Industrial and Applied Mathematics, 42 (2000), 599. Google Scholar [5] C. Bowman, A. B. Gumel, J. Wu, P. van den Driessche and H. Zhu, A mathematical model for assessing control strategies against West Nile virus,, Bull. Math. Biol., 67 (2005), 1107. doi: 10.1016/j.bulm.2005.01.002. Google Scholar [6] Miranda I. Teboh-Ewungkem, Chandra N. Podder and Abba B. Gumel, Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics,, Bulletin of Mathematical Biology, 72 (2010), 63. doi: 10.1007/s11538-009-9437-3. Google Scholar [7] Rebecca Culshaw, Shigui Ruan and Raymond J. Spiteri, Optimal HIV treatment by maximising immune response,, J. Math. Biol., 48 (2004), 545. doi: 10.1007/s00285-003-0245-3. Google Scholar [8] Xiefei Yan and Yun Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics,, Mathematical and Computer Modelling, 47 (2008), 235. doi: 10.1016/j.mcm.2007.04.003. Google Scholar [9] Roberto C. A. Thomé, Hyun Mo Yang and Lourdes Esteva, Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide,, Mathematical Biosciences, 223 (2010), 12. doi: 10.1016/j.mbs.2009.08.009. Google Scholar [10] Kbenesh W. Blayneh, Abba B. Gumel, Suzanne Lenhart and Tim Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus,, Bulletin of Mathematical Biology, 72 (2010), 1006. doi: 10.1007/s11538-009-9480-0. Google Scholar [11] 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 [12] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar [13] Nakul Chitnis, James M. Hyman and Jim 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. doi: 10.1007/s11538-008-9299-0. Google Scholar [14] 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 [15] W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Applications of Mathematics, (1975). Google Scholar [16] Hem Raj Joshi, Optimal control of an HIV immunology model,, Optim. Control Appl. Meth., 23 (2002), 199. doi: 10.1002/oca.710. Google Scholar [17] S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007). Google Scholar [18] Thomas Smith, Gerry F. Killeen, Nicolas Maire, Amanda Ross, Louis Molineaux, Fabrizio Tediosi, Guy Hutton, Jürg Utzinger, Klaus Dietz and Marcel 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. Google Scholar [19] , "Roll Back Malaria Partnership, 2005," RBM World Malaria Report 2005,, 2003. Available from: \url{http://rbm.who.int/wmr2005/index.html}., (). Google Scholar [20] K. Renee Fister, Suzanne Lenhart and Joseph Scott Mcnally, Optimizing chemotherapy in an HIV model,, Electronic Journal of Differential Equations, 1998 (). Google Scholar [21] W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations,, Computing, 20 (1978), 229. doi: 10.1007/BF02251947. Google Scholar

show all references

##### References:
 [1] World Health Organization Expert Committee on Malaria, 20th Report,, WHO Regional Office for Africa, (2003). Google Scholar [2] S. W. Lindsay, W. J. M. Martens, Malaria in the African highlands: Past, present and future,, Bull. WHO, 76 (1998), 33. Google Scholar [3] G. Zhou, N. Minakawa, A. K. Githeko and G. Yan, Association between climate variability and malaria epidemics in the east African highlands,, Proc. Natl. Acad. Sci., 101 (2004), 2375. doi: 10.1073/pnas.0308714100. Google Scholar [4] Herbert W. Hethcote, The mathematics of infectious diseases,, Socity for Industrial and Applied Mathematics, 42 (2000), 599. Google Scholar [5] C. Bowman, A. B. Gumel, J. Wu, P. van den Driessche and H. Zhu, A mathematical model for assessing control strategies against West Nile virus,, Bull. Math. Biol., 67 (2005), 1107. doi: 10.1016/j.bulm.2005.01.002. Google Scholar [6] Miranda I. Teboh-Ewungkem, Chandra N. Podder and Abba B. Gumel, Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics,, Bulletin of Mathematical Biology, 72 (2010), 63. doi: 10.1007/s11538-009-9437-3. Google Scholar [7] Rebecca Culshaw, Shigui Ruan and Raymond J. Spiteri, Optimal HIV treatment by maximising immune response,, J. Math. Biol., 48 (2004), 545. doi: 10.1007/s00285-003-0245-3. Google Scholar [8] Xiefei Yan and Yun Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics,, Mathematical and Computer Modelling, 47 (2008), 235. doi: 10.1016/j.mcm.2007.04.003. Google Scholar [9] Roberto C. A. Thomé, Hyun Mo Yang and Lourdes Esteva, Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide,, Mathematical Biosciences, 223 (2010), 12. doi: 10.1016/j.mbs.2009.08.009. Google Scholar [10] Kbenesh W. Blayneh, Abba B. Gumel, Suzanne Lenhart and Tim Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus,, Bulletin of Mathematical Biology, 72 (2010), 1006. doi: 10.1007/s11538-009-9480-0. Google Scholar [11] 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 [12] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar [13] Nakul Chitnis, James M. Hyman and Jim 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. doi: 10.1007/s11538-008-9299-0. Google Scholar [14] 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 [15] W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Applications of Mathematics, (1975). Google Scholar [16] Hem Raj Joshi, Optimal control of an HIV immunology model,, Optim. Control Appl. Meth., 23 (2002), 199. doi: 10.1002/oca.710. Google Scholar [17] S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007). Google Scholar [18] Thomas Smith, Gerry F. Killeen, Nicolas Maire, Amanda Ross, Louis Molineaux, Fabrizio Tediosi, Guy Hutton, Jürg Utzinger, Klaus Dietz and Marcel 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. Google Scholar [19] , "Roll Back Malaria Partnership, 2005," RBM World Malaria Report 2005,, 2003. Available from: \url{http://rbm.who.int/wmr2005/index.html}., (). Google Scholar [20] K. Renee Fister, Suzanne Lenhart and Joseph Scott Mcnally, Optimizing chemotherapy in an HIV model,, Electronic Journal of Differential Equations, 1998 (). Google Scholar [21] W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations,, Computing, 20 (1978), 229. doi: 10.1007/BF02251947. Google Scholar
 [1] Sanjukta Hota, Folashade Agusto, Hem Raj Joshi, Suzanne Lenhart. Optimal control and stability analysis of an epidemic model with education campaign and treatment. Conference Publications, 2015, 2015 (special) : 621-634. doi: 10.3934/proc.2015.0621 [2] Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241 [3] John T. Betts, Stephen L. Campbell, Claire Digirolamo. Initial guess sensitivity in Computational optimal control problems. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019031 [4] Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Optimal control for an epidemic in populations of varying size. Conference Publications, 2015, 2015 (special) : 549-561. doi: 10.3934/proc.2015.0549 [5] Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045 [6] Ellina Grigorieva, Evgenii Khailov. Determination of the optimal controls for an Ebola epidemic model. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1071-1101. doi: 10.3934/dcdss.2018062 [7] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 [8] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [9] Yanqin Bai, Yudan Wei, Qian Li. An optimal trade-off model for portfolio selection with sensitivity of parameters. Journal of Industrial & Management Optimization, 2017, 13 (2) : 947-965. doi: 10.3934/jimo.2016055 [10] Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030 [11] Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021 [12] Heinz Schättler, Urszula Ledzewicz. Fields of extremals and sensitivity analysis for multi-input bilinear optimal control problems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4611-4638. doi: 10.3934/dcds.2015.35.4611 [13] Arti Mishra, Benjamin Ambrosio, Sunita Gakkhar, M. A. Aziz-Alaoui. A network model for control of dengue epidemic using sterile insect technique. Mathematical Biosciences & Engineering, 2018, 15 (2) : 441-460. doi: 10.3934/mbe.2018020 [14] Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729 [15] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [16] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [17] Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457 [18] Enrique Fernández-Cara, Juan Límaco, Laurent Prouvée. Optimal control of a two-equation model of radiotherapy. Mathematical Control & Related Fields, 2018, 8 (1) : 117-133. doi: 10.3934/mcrf.2018005 [19] Erika Asano, Louis J. Gross, Suzanne Lenhart, Leslie A. Real. Optimal control of vaccine distribution in a rabies metapopulation model. Mathematical Biosciences & Engineering, 2008, 5 (2) : 219-238. doi: 10.3934/mbe.2008.5.219 [20] Huaying Guo, Jin Liang. An optimal control model of carbon reduction and trading. Mathematical Control & Related Fields, 2016, 6 (4) : 535-550. doi: 10.3934/mcrf.2016015

2018 Impact Factor: 1.292