American Institute of Mathematical Sciences

Advanced
2011, 8(1): 141-170. doi: 10.3934/mbe.2011.8.141

Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza

Olivia Prosper 1, , Omar Saucedo 2, , Doria Thompson 3, , Griselle Torres-Garcia 4, , Xiaohong Wang 4, and  Carlos Castillo-Chavez 5,

1. 

Department of Mathematics, University of Florida, Gainesville, FL 32611, United States
2. 

Department of Mathematics, Texas A&M University, College Station, TX 77843, United States
3. 

Department of Mathematics, Spelman College, Atlanta, GA 30314, United States
4. 

School of Human Evolution and Social Change, Mathematical, Computational and Modeling Science Center, Arizona State University, Tempe, AZ 85287, United States, United States
5. 

Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287

Received  June 2010 Revised  September 2010 Published  January 2011

The lessons learned from the 2009-2010 H1N1 influenza pandemic, as it moves out of the limelight, should not be under-estimated, particularly since the probability of novel influenza epidemics in the near future is not negligible and the potential consequences might be huge. Hence, as the world, particularly the industrialized world, responded to the potentially devastating effects of this novel A-H1N1 strain with substantial resources, reminders of the recurrent loss of life from a well established foe, seasonal influenza, could not be ignored. The uncertainties associated with the reported and expected levels of morbidity and mortality with this novel A-H1N1 live in a backdrop of $36,000$ deaths, over 200,000 hospitalizations, and millions of infections (20% of the population) attributed to seasonal influenza in the USA alone, each year. So, as the Northern Hemisphere braced for the possibility of a potentially "lethal" second wave of the novel A-H1N1 without a vaccine ready to mitigate its impact, questions of who should be vaccinated first if a vaccine became available, came to the forefront of the discussion. Uncertainty grew as we learned that the vaccine, once available, would be unevenly distributed around the world. Nations capable of acquiring large vaccine supplies soon became aware that those who could pay would have to compete for a limited vaccine stockpile. The challenges faced by nations dealing jointly with seasonal and novel A-H1N1 co-circulating strains under limited resources, that is, those with no access to novel A-H1N1 vaccine supplies, limited access to the seasonal influenza vaccine, and limited access to antivirals (like Tamiflu) are explored in this study. One- and two-strain models are introduced to mimic the influenza dynamics of a single and co-circulating strains, in the context of a single epidemic outbreak. Optimal control theory is used to identify and evaluate the "best" control policies. The controls account for the cost associated with social distancing and antiviral treatment policies. The optimal policies identified might have, if implemented, a substantial impact on the novel H1N1 and seasonal influenza co-circulating dynamics. Specifically, the implementation of antiviral treatment might reduce the number of influenza cases by up to 60% under a reasonable seasonal vaccination strategy, but only by up to 37% when the seasonal vaccine is not available. Optimal social distancing policies alone can be as effective as the combination of multiple policies, reducing the total number of influenza cases by more than 99% within a single outbreak, an unrealistic but theoretically possible outcome for isolated populations with limited resources.
Keywords: Control Theory, Seasonal Influenza, Basic Reproductive Number..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Olivia Prosper, Omar Saucedo, Doria Thompson, Griselle Torres-Garcia, Xiaohong Wang, Carlos Castillo-Chavez. Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza. Mathematical Biosciences & Engineering, 2011, 8 (1) : 141-170. doi: 10.3934/mbe.2011.8.141
References:
[1]

L. Altman, "Many Swine Flu Cases Have no Fever," New York Times, 2009. Google Scholar

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford University Press, Oxford, 1991. Google Scholar

[3]

I. G. Barr, J. McCauley, N. Cox, R. Daniels, O. G. Engelhardt, K. Fukuda, G. Grohmann, A. Hay, A. Kelso, A. Klimov, T. Odagiri, D. Smith, C. Russell, M. Tashiro, R. Webby, J. Wood, Z. Ye and W. Zhang, Epidemiological, antigenic and genetic characteristics of seasonal influenzaA(H1N1), A(H3N2) and B influenza viruses: Basis for the WHO recommendation on the composition of influenza vaccines for use in the 2009–2010 Northern Hemisphere season, Vaccine, 28 (2010), 1156-1167. doi: 10.1016/j.vaccine.2009.11.043.  Google Scholar

[4]

H. Behnke, Optimal control of deterministic epidemics, Optimal Control Application Methods, 21 (2000), 269-285. doi: 10.1002/oca.678.  Google Scholar

[5]

W. I. B. Beveridge, "Influenza: The Last Great Plague. An Unfinished Story of Discovery," Prodist, 1977. Google Scholar

[6]

F. Brauer, Z. Feng and C. Castillo-Chavez, Discrete epidemic models, Mathematical Biosciences and Engineering, 7 (2010), 1-15. doi: 10.3934/mbe.2010.7.1.  Google Scholar

[7]

C. Castillo-Chavez, H. Hethcote, V. Andreason, S. A. Levin and W. M. Liu, Cross-immunity in the dynamics of homogeneous and heterogeneous populations, Mathematical Ecology, (1988), 303-316. Google Scholar

[8]

Centers for Disease Control and Prevention (CDC), Key facts about seasonal influenza,, \url{http://www.cdc.gov/flu/keyfacts.htm}., ().   Google Scholar

[9]

Centers for Disease Control and Prevention (CDC), Monitoring influenza activity, including 2009 H1N1, (2009), Monitoring Influenza Activity, Including 2009 h1n1. Google Scholar

[10]

Centers for Disease Control and Prevention (CDC), Serum cross-reactive antibody response to a novel influenza A(H1N1) virus after vaccination with seasonal influenza vaccine, MMWR Morb Mortal Wkly Rep, 58 (2009), 521-524. Google Scholar

[11]

G. Chowell, M. A. Miller and C. Viboud, Seasonal influenza in the United States, France, and Australia: Transmission an prospects for control, Epidem. Infect., 136 (2008), 852-864. doi: 10.1017/S0950268807009144.  Google Scholar

[12]

G. Chowell, S. M. Bertozzi, M. A. Colchero, H. Lopez-Gatell, C. Alpuche-Aranda, M. Hernandez and M. A. Miller, Severe respiratory disease concurrent with the circulation of H1N1 influenza, The New England Journal of Medicine, 361 (2009), 674-679. doi: 10.1056/NEJMoa0904023.  Google Scholar

[13]

Brian Coburn, "Multi-species Influenza Models with Recombination," Ph.D thesis, University of Miami in Coral Gables,FL, 2009.  Google Scholar

[14]

R. Couch and J. Kasel, Immunity to influenza in man, Annual Reviews in Microbiology, 37 (2002), 529-549. doi: 10.1146/annurev.mi.37.100183.002525.  Google Scholar

[15]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," John Wiley & Sons, Ltd., Chichester, 2000.  Google Scholar

[16]

D. J. D. Earn, J. Dushoff and S. A. Levin, Ecology and evolution of the flu, Trends Ecol. Evol., 17 (2002), 334-340. doi: 10.1016/S0169-5347(02)02502-8.  Google Scholar

[17]

S. Echevarría-Zuno, J. M. Mejía-Aranguré, A. V. Mar-Obeso, C. Grajales-Muñiz, E. Robles-Pérez, M. González-León, M. C. Ortega-Alvarez, C. Gonzalez-Bonilla, R. A. Rascón-Pacheco and V. H. Borja-Aburto, Infection and death from influenza A H1N1 virus in Mexico: A retrospective analysis, Lancet, 374 (2009), 2072-2079. doi: 10.1016/S0140-6736(09)61638-X.  Google Scholar

[18]

A. Esteves-Jaramillo, S. B. Omer and E. Gonzalez-Diaz, Acceptance of a vaccine against novel influenza A (H1N1) virus among health care workers in two major cities in Mexico, Archives of Medical Research, 40 (2009), 705-711. doi: 10.1016/j.arcmed.2010.01.004.  Google Scholar

[19]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochasitic Optimal Control," Springer-Verlag, New York, 1994. Google Scholar

[20]

FLU. GOV, 2009 H1N1 vaccine doses allocated, ordered, and shipped by project area, (2010),, \url{http://www.flu.gov/individualfamily/vaccination/supply.html}., ().   Google Scholar

[21]

M. A. Herrera-Valdez, M. Cruz-Aponte and C. Castillo-Chavez, Multiple waves for the same pandemic: Local transportation and social distancing explain the dynamics of the A/H1N1 epidemic during 2009 in Mexico, (2010). Google Scholar

[22]

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

[23]

P. Y. Lee, D. B. Matchar, D. A. Clements, J. Huber, J. D. Hamilton and E. D. Peterson, Economic analysis of influenza vaccination and antiviral treatment for healthy working adults, Ann. Intern. Med., 137 (2002), 225-231. Google Scholar

[24]

S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control of influenza pandemics: the role of antiviral treatment and isolation, Journal of Theoretical Biology, 265 (2010), 136-150. doi: 10.1016/j.jtbi.2010.04.003.  Google Scholar

[25]

S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, 2007. Google Scholar

[26]

E. Malkin, Flu? What flu?,, The New York Times, ().   Google Scholar

[27]

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, Eurosurveillance, 14 (2009), 1-4. Google Scholar

[28]

M. Nuno, G. Chowell and A. B. Gumel, Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK and the Netherlands, Journal of The Royal Society Interface, 4 (2007), 505-521. doi: 10.1098/rsif.2006.0186.  Google Scholar

[29]

J. Plotkin, J. Dushoff and S. Levin, Hemagglutinin sequence clusters and the antigenic evolution of influenza A virus, Proceedings of the National Academy of Sciences, 99 (2002), 6263-6268. doi: 10.1073/pnas.082110799.  Google Scholar

[30]

L. S. Pontryagin, R. V. Boltyanski, R. V. Gamkrelidge and E. F. Mischenko, "The Mathematical Theory of Optimal Processes," John Wiley and Sons, N.Y., 1962.  Google Scholar

[31]

Prevent Influenza Now! Sponsored by the National Influenza Vaccine Summit, Influenza vaccine availability tracking system (IVATS),, \url{http://www.preventinfluenza.org/ivats/}., ().   Google Scholar

[32]

C. E. Shoichet, Mexico still waiting for most swine flu vaccines, (2010), http://www.boston.com/business/articles/2010/01/13/mexico_still_waiting_for_most_swine _flu_vaccines. Google Scholar

[33]

E. Spackman, D. Stallknecht, R. Slemons, K. Winker, D. L. Suarez, M. Scott and D. E. Swayne, Phylogenetic analyses of type A influenza genes in natural reservoir species in North America reveals genetic variation, Virus research, 114 (2005), 89-100. doi: 10.1016/j.virusres.2005.05.013.  Google Scholar

[34]

R. Stengel, Optimal control and estimation,, http://www.princeton.edu/ stengel/MAE546.html., ().   Google Scholar

[35]

T. Suess, U. Buchholz, S. Dupke, R. Grunow, M. an der Heiden, A. Heider, B. Biere, B. Schweiger, W. Haas and G. Krause, Shedding and transmission of novel influenza virus A/H1N1 infection in households—Germany, 2009, American Journal of Epidemiology, 171 (2010), 1157-1164. doi: 10.1093/aje/kwq071.  Google Scholar

[36]

J. K. Taubenberger, D. M. Morens, 1918 influenza: the mother of all pandemics, Emerging Infectious Diseases, (2009), http://www.cdc.gov/ncidod/EID/vol12no01/05-0979.htm. Google Scholar

[37]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[38]

World Health Organization, Recommended composition of influenza of influenza virus vaccines for use in the 2001-2002 season, Wkly. Epidemiol. Rec., 76 (2001), 58-61. Google Scholar

show all references

References:
[1]

L. Altman, "Many Swine Flu Cases Have no Fever," New York Times, 2009. Google Scholar

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford University Press, Oxford, 1991. Google Scholar

[3]

I. G. Barr, J. McCauley, N. Cox, R. Daniels, O. G. Engelhardt, K. Fukuda, G. Grohmann, A. Hay, A. Kelso, A. Klimov, T. Odagiri, D. Smith, C. Russell, M. Tashiro, R. Webby, J. Wood, Z. Ye and W. Zhang, Epidemiological, antigenic and genetic characteristics of seasonal influenzaA(H1N1), A(H3N2) and B influenza viruses: Basis for the WHO recommendation on the composition of influenza vaccines for use in the 2009–2010 Northern Hemisphere season, Vaccine, 28 (2010), 1156-1167. doi: 10.1016/j.vaccine.2009.11.043.  Google Scholar

[4]

H. Behnke, Optimal control of deterministic epidemics, Optimal Control Application Methods, 21 (2000), 269-285. doi: 10.1002/oca.678.  Google Scholar

[5]

W. I. B. Beveridge, "Influenza: The Last Great Plague. An Unfinished Story of Discovery," Prodist, 1977. Google Scholar

[6]

F. Brauer, Z. Feng and C. Castillo-Chavez, Discrete epidemic models, Mathematical Biosciences and Engineering, 7 (2010), 1-15. doi: 10.3934/mbe.2010.7.1.  Google Scholar

[7]

C. Castillo-Chavez, H. Hethcote, V. Andreason, S. A. Levin and W. M. Liu, Cross-immunity in the dynamics of homogeneous and heterogeneous populations, Mathematical Ecology, (1988), 303-316. Google Scholar

[8]

Centers for Disease Control and Prevention (CDC), Key facts about seasonal influenza,, \url{http://www.cdc.gov/flu/keyfacts.htm}., ().   Google Scholar

[9]

Centers for Disease Control and Prevention (CDC), Monitoring influenza activity, including 2009 H1N1, (2009), Monitoring Influenza Activity, Including 2009 h1n1. Google Scholar

[10]

Centers for Disease Control and Prevention (CDC), Serum cross-reactive antibody response to a novel influenza A(H1N1) virus after vaccination with seasonal influenza vaccine, MMWR Morb Mortal Wkly Rep, 58 (2009), 521-524. Google Scholar

[11]

G. Chowell, M. A. Miller and C. Viboud, Seasonal influenza in the United States, France, and Australia: Transmission an prospects for control, Epidem. Infect., 136 (2008), 852-864. doi: 10.1017/S0950268807009144.  Google Scholar

[12]

G. Chowell, S. M. Bertozzi, M. A. Colchero, H. Lopez-Gatell, C. Alpuche-Aranda, M. Hernandez and M. A. Miller, Severe respiratory disease concurrent with the circulation of H1N1 influenza, The New England Journal of Medicine, 361 (2009), 674-679. doi: 10.1056/NEJMoa0904023.  Google Scholar

[13]

Brian Coburn, "Multi-species Influenza Models with Recombination," Ph.D thesis, University of Miami in Coral Gables,FL, 2009.  Google Scholar

[14]

R. Couch and J. Kasel, Immunity to influenza in man, Annual Reviews in Microbiology, 37 (2002), 529-549. doi: 10.1146/annurev.mi.37.100183.002525.  Google Scholar

[15]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," John Wiley & Sons, Ltd., Chichester, 2000.  Google Scholar

[16]

D. J. D. Earn, J. Dushoff and S. A. Levin, Ecology and evolution of the flu, Trends Ecol. Evol., 17 (2002), 334-340. doi: 10.1016/S0169-5347(02)02502-8.  Google Scholar

[17]

S. Echevarría-Zuno, J. M. Mejía-Aranguré, A. V. Mar-Obeso, C. Grajales-Muñiz, E. Robles-Pérez, M. González-León, M. C. Ortega-Alvarez, C. Gonzalez-Bonilla, R. A. Rascón-Pacheco and V. H. Borja-Aburto, Infection and death from influenza A H1N1 virus in Mexico: A retrospective analysis, Lancet, 374 (2009), 2072-2079. doi: 10.1016/S0140-6736(09)61638-X.  Google Scholar

[18]

A. Esteves-Jaramillo, S. B. Omer and E. Gonzalez-Diaz, Acceptance of a vaccine against novel influenza A (H1N1) virus among health care workers in two major cities in Mexico, Archives of Medical Research, 40 (2009), 705-711. doi: 10.1016/j.arcmed.2010.01.004.  Google Scholar

[19]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochasitic Optimal Control," Springer-Verlag, New York, 1994. Google Scholar

[20]

FLU. GOV, 2009 H1N1 vaccine doses allocated, ordered, and shipped by project area, (2010),, \url{http://www.flu.gov/individualfamily/vaccination/supply.html}., ().   Google Scholar

[21]

M. A. Herrera-Valdez, M. Cruz-Aponte and C. Castillo-Chavez, Multiple waves for the same pandemic: Local transportation and social distancing explain the dynamics of the A/H1N1 epidemic during 2009 in Mexico, (2010). Google Scholar

[22]

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

[23]

P. Y. Lee, D. B. Matchar, D. A. Clements, J. Huber, J. D. Hamilton and E. D. Peterson, Economic analysis of influenza vaccination and antiviral treatment for healthy working adults, Ann. Intern. Med., 137 (2002), 225-231. Google Scholar

[24]

S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control of influenza pandemics: the role of antiviral treatment and isolation, Journal of Theoretical Biology, 265 (2010), 136-150. doi: 10.1016/j.jtbi.2010.04.003.  Google Scholar

[25]

S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, 2007. Google Scholar

[26]

E. Malkin, Flu? What flu?,, The New York Times, ().   Google Scholar

[27]

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, Eurosurveillance, 14 (2009), 1-4. Google Scholar

[28]

M. Nuno, G. Chowell and A. B. Gumel, Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic influenza: Scenarios for the US, UK and the Netherlands, Journal of The Royal Society Interface, 4 (2007), 505-521. doi: 10.1098/rsif.2006.0186.  Google Scholar

[29]

J. Plotkin, J. Dushoff and S. Levin, Hemagglutinin sequence clusters and the antigenic evolution of influenza A virus, Proceedings of the National Academy of Sciences, 99 (2002), 6263-6268. doi: 10.1073/pnas.082110799.  Google Scholar

[30]

L. S. Pontryagin, R. V. Boltyanski, R. V. Gamkrelidge and E. F. Mischenko, "The Mathematical Theory of Optimal Processes," John Wiley and Sons, N.Y., 1962.  Google Scholar

[31]

Prevent Influenza Now! Sponsored by the National Influenza Vaccine Summit, Influenza vaccine availability tracking system (IVATS),, \url{http://www.preventinfluenza.org/ivats/}., ().   Google Scholar

[32]

C. E. Shoichet, Mexico still waiting for most swine flu vaccines, (2010), http://www.boston.com/business/articles/2010/01/13/mexico_still_waiting_for_most_swine _flu_vaccines. Google Scholar

[33]

E. Spackman, D. Stallknecht, R. Slemons, K. Winker, D. L. Suarez, M. Scott and D. E. Swayne, Phylogenetic analyses of type A influenza genes in natural reservoir species in North America reveals genetic variation, Virus research, 114 (2005), 89-100. doi: 10.1016/j.virusres.2005.05.013.  Google Scholar

[34]

R. Stengel, Optimal control and estimation,, http://www.princeton.edu/ stengel/MAE546.html., ().   Google Scholar

[35]

T. Suess, U. Buchholz, S. Dupke, R. Grunow, M. an der Heiden, A. Heider, B. Biere, B. Schweiger, W. Haas and G. Krause, Shedding and transmission of novel influenza virus A/H1N1 infection in households—Germany, 2009, American Journal of Epidemiology, 171 (2010), 1157-1164. doi: 10.1093/aje/kwq071.  Google Scholar

[36]

J. K. Taubenberger, D. M. Morens, 1918 influenza: the mother of all pandemics, Emerging Infectious Diseases, (2009), http://www.cdc.gov/ncidod/EID/vol12no01/05-0979.htm. Google Scholar

[37]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[38]

World Health Organization, Recommended composition of influenza of influenza virus vaccines for use in the 2001-2002 season, Wkly. Epidemiol. Rec., 76 (2001), 58-61. Google Scholar

[1]

Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377
[2]

Oren Barnea, Rami Yaari, Guy Katriel, Lewi Stone. Modelling seasonal influenza in Israel. Mathematical Biosciences & Engineering, 2011, 8 (2) : 561-573. doi: 10.3934/mbe.2011.8.561
[3]

Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks. The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences & Engineering, 2009, 6 (2) : 261-282. doi: 10.3934/mbe.2009.6.261
[4]

Pierre Magal, Ahmed Noussair, Glenn Webb, Yixiang Wu. Modeling epidemic outbreaks in geographical regions: Seasonal influenza in Puerto Rico. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3535-3550. doi: 10.3934/dcdss.2020237
[5]

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
[6]

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
[7]

Sherry Towers, Katia Vogt Geisse, Chia-Chun Tsai, Qing Han, Zhilan Feng. The impact of school closures on pandemic influenza: Assessing potential repercussions using a seasonal SIR model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 413-430. doi: 10.3934/mbe.2012.9.413
[8]

Pierre Magal, Glenn F. Webb, Yixiang Wu. Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2185-2202. doi: 10.3934/dcdsb.2019223
[9]

Shuang-Lin Jing, Hai-Feng Huo, Hong Xiang. Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021113
[10]

Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056
[11]

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
[12]

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
[13]

Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170
[14]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437
[15]

E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
[16]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[17]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1091-1116. doi: 10.3934/dcdsb.2004.4.1091
[18]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
[19]

Ting Kang, Qimin Zhang, Haiyan Wang. Optimal control of an avian influenza model with multiple time delays in state and control variables. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4147-4171. doi: 10.3934/dcdsb.2020278
[20]

Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (133)
  • HTML views (0)
  • Cited by (26)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy