February  2018, 15(1): 209-232. doi: 10.3934/mbe.2018009

Prediction of influenza peaks in Russian cities: Comparing the accuracy of two SEIR models

ITMO University, 49 Kronverksky Pr, 197101, St. Petersburg, Russia

* Corresponding author: vnleonenko@yandex.ru

Received  November 08, 2016 Accepted  April 06, 2017 Published  May 2017

This paper is dedicated to the application of two types of SEIR models to the influenza outbreak peak prediction in Russian cities. The first one is a continuous SEIR model described by a system of ordinary differential equations. The second one is a discrete model formulated as a set of difference equations, which was used in the Baroyan-Rvachev modeling framework for the influenza outbreak prediction in the Soviet Union. The outbreak peak day and height predictions were performed by calibrating both models to varied-size samples of long-term data on ARI incidence in Moscow, Saint Petersburg, and Novosibirsk. The accuracy of the modeling predictions on incomplete data was compared with a number of other peak forecasting methods tested on the same dataset. The drawbacks of the described prediction approach and possible ways to overcome them are discussed.

Citation: Vasiliy N. Leonenko, Sergey V. Ivanov. Prediction of influenza peaks in Russian cities: Comparing the accuracy of two SEIR models. Mathematical Biosciences & Engineering, 2018, 15 (1) : 209-232. doi: 10.3934/mbe.2018009
References:
[1]

O. Baroyan, U. Basilevsky, V. Ermakov, K. Frank, L. Rvachev and V. Shashkov, Computer modelling of influenza epidemics for large-scale systems of cities and territories, in Proc. WHO Symposium on Quantitative Epidemiology, Moscow, 1970.Google Scholar

[2]

R. BurgerG. ChowellP. Mulet and L. Villada, Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile, Mathematical Biosciences and Engineering, 13 (2016), 43-65. doi: 10.3934/mbe.2016.13.43. Google Scholar

[3]

CDC, Influenza signs and symptoms and the role of laboratory diagnostics, [online], http://www.cdc.gov/flu/professionals/diagnosis/labrolesprocedures.htm.Google Scholar

[4]

CDC, People with heart disease and those who have had a stroke are at high risk of developing complications from influenza (the flu), [online], http://www.cdc.gov/flu/heartdisease/.Google Scholar

[5]

J. -P. Chretien, D. George, J. Shaman, R. A. Chitale and F. E. McKenzie, Influenza forecasting in human populations: A scoping review, PloS one, 9 (2014), e94130. doi: 10.1371/journal.pone.0094130. Google Scholar

[6]

A. D. Cliff, P. Haggett and J. K. Ord, Spatial Aspects of Influenza Epidemics, Routledge, 1986. Google Scholar

[7]

V. Colizza, A. Barrat, M. Barthelemy, A. -J. Valleron and A. Vespignani, Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions, PLoS Med, 4 (2007), e13.Google Scholar

[8]

S. Cook, C. Conrad, A. L. Fowlkes and M. H. Mohebbi, Assessing google flu trends performance in the united states during the 2009 influenza virus a (h1n1) pandemic PloS one, 6 (2011), e23610. doi: 10.1371/journal.pone.0023610. Google Scholar

[9]

N. Goeyvaerts, L. Willem, K. Van~Kerckhove, Y. Vandendijck, G. Hanquet, P. Beutels and N. Hens, Estimating dynamic transmission model parameters for seasonal influenza by fitting to age and season-specific influenza-like illness incidence Epidemics, 13 (2015), p1. doi: 10.1016/j.epidem.2015.04.002. Google Scholar

[10]

I. HallR. GaniH. Hughes and S. Leach, Real-time epidemic forecasting for pandemic influenza, Epidemiology and Infection, 135 (2007), 372-385. doi: 10.1017/S0950268806007084. Google Scholar

[11]

D. He, J. Dushoff, R. Eftimie and D. J. Earn, Patterns of spread of influenza A in Canada, Proceedings of the Royal Society of London B: Biological Sciences, 280 (2013), 20131174. doi: 10.1098/rspb.2013.1174. Google Scholar

[12]

K. S. HickmannG. FairchildR. PriedhorskyN. GenerousJ. M. HymanA. Deshpande and S. Y. Del Valle, Forecasting the 2013-2014 influenza season using wikipedia, PLoS Comput Biol, 11 (2015), e1004239. doi: 10.1371/journal.pcbi.1004239. Google Scholar

[13]

A. Hyder, D. L. Buckeridge and B. Leung, Predictive validation of an influenza spread model PloS one, 8 (2013), e65459. doi: 10.1371/journal.pone.0065459. Google Scholar

[14]

F. Institute, Research Institute of Influenza website, [online], http://influenza.spb.ru/en/.Google Scholar

[15]

Y. G. Ivannikov and A. T. Ismagulov, Epidemiologiya Grippa (The Epidemiology of Influenza), Almaty, Kazakhstan, 1983, In Russian.Google Scholar

[16]

Y. Ivannikov and P. Ogarkov, An experience of mathematical computing forecasting of the influenza epidemics for big territory, Journal of Infectology, 4 (2012), 101-106. Google Scholar

[17]

V. N. Leonenko and S. V. Ivanov, Fitting the SEIR model of seasonal influenza outbreak to the incidence data for Russian cities, Russian Journal of Numerical Analysis and Mathematical Modelling, 31 (2016), 267-279. doi: 10.1515/rnam-2016-0026. Google Scholar

[18]

V. N. LeonenkoS. V. Ivanov and Y. K. Novoselova, A computational approach to investigate patterns of acute respiratory illness dynamics in the regions with distinct seasonal climate transitions, Procedia Computer Science, 80 (2016), 2402-2412. doi: 10.1016/j.procs.2016.05.538. Google Scholar

[19]

V. N. LeonenkoY. K. Novoselova and K. M. Ong, Influenza outbreaks forecasting in Russian cities: Is Baroyan-Rvachev approach still applicable?, Procedia Computer Science, 101 (2016), 282-291. doi: 10.1016/j.procs.2016.11.033. Google Scholar

[20]

V. N. LeonenkoN. V. Pertsev and M. Artzrouni, Using high performance algorithms for the hybrid simulation of disease dynamics on CPU and GPU, Procedia Computer Science, 51 (2015), 150-159. doi: 10.1016/j.procs.2015.05.214. Google Scholar

[21]

D. C. Liu and J. Nocedal, On the limited memory bfgs method for large scale optimization, Mathematical programming, 45 (1989), 503-528. doi: 10.1007/BF01589116. Google Scholar

[22]

A. RomanyukhaT. Sannikova and I. Drynov, The origin of acute respiratory epidemics, Herald of the Russian Academy of Sciences, 81 (2011), 31-34. doi: 10.1134/S1019331611010114. Google Scholar

[23]

L. A. Rvachev and I. M. Longini, A mathematical model for the global spread of influenza, Mathematical Biosciences, 75 (1985), 1-22. doi: 10.1016/0025-5564(85)90063-X. Google Scholar

[24]

J. Shaman, V. E. Pitzer, C. Viboud, B. T. Grenfell and M. Lipsitch, Absolute humidity and the seasonal onset of influenza in the continental United States, PLoS Biol, 8 (2010), e1000316.Google Scholar

[25]

J. TameriusM. I. NelsonS. Z. ZhouC. ViboudM. A. Miller and W. J. Alonso, Global influenza seasonality: Reconciling patterns across temperate and tropical regions, Environmental Health Perspectives, 119 (2011), 439-445. doi: 10.1289/ehp.1002383. Google Scholar

[26]

J. TruscottC. FraserS. CauchemezA. MeeyaiW. HinsleyC. A. DonnellyA. Ghani and N. Ferguson, Essential epidemiological mechanisms underpinning the transmission dynamics of seasonal influenza, Journal of The Royal Society Interface, 9 (2011), 304-312. doi: 10.1098/rsif.2011.0309. Google Scholar

[27]

S. P. van NoortR. ÁguasS. Ballesteros and M. G. M. Gomes, The role of weather on the relation between influenza and influenza-like illness, Journal of Theoretical Biology, 298 (2012), 131-137. Google Scholar

[28]

C. ViboudO. N. BjornstadD. L. SmithL. SimonsenM. A. Miller and B. T. Grenfell, Synchrony, waves, and spatial hierarchies in the spread of influenza, Science, 312 (2006), 447-451. doi: 10.1126/science.1125237. Google Scholar

[29]

WHO, Influenza (seasonal). Fact sheet No. 211, March 2014. , [online], http://www.who.int/mediacentre/factsheets/fs211/en/.Google Scholar

[30]

WHO, Surveillance case definitions for ILI and SARI, [online], http://www.who.int/influenza/surveillance_monitoring/ili_sari_surveillance_case_definition/en/.Google Scholar

[31]

R. Yaari, G. Katriel, A. Huppert, J. Axelsen and L. Stone, Modelling seasonal influenza: The role of weather and punctuated antigenic drift, Journal of The Royal Society Interface, 10 (2013), 20130298. doi: 10.1098/rsif.2013.0298. Google Scholar

[32]

W. Yang, B. J. Cowling, E. H. Lau and J. Shaman, Forecasting influenza epidemics in hong kong, PLoS Comput Biol, 11 (2015), e1004383. doi: 10.1371/journal.pcbi.1004383. Google Scholar

show all references

References:
[1]

O. Baroyan, U. Basilevsky, V. Ermakov, K. Frank, L. Rvachev and V. Shashkov, Computer modelling of influenza epidemics for large-scale systems of cities and territories, in Proc. WHO Symposium on Quantitative Epidemiology, Moscow, 1970.Google Scholar

[2]

R. BurgerG. ChowellP. Mulet and L. Villada, Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile, Mathematical Biosciences and Engineering, 13 (2016), 43-65. doi: 10.3934/mbe.2016.13.43. Google Scholar

[3]

CDC, Influenza signs and symptoms and the role of laboratory diagnostics, [online], http://www.cdc.gov/flu/professionals/diagnosis/labrolesprocedures.htm.Google Scholar

[4]

CDC, People with heart disease and those who have had a stroke are at high risk of developing complications from influenza (the flu), [online], http://www.cdc.gov/flu/heartdisease/.Google Scholar

[5]

J. -P. Chretien, D. George, J. Shaman, R. A. Chitale and F. E. McKenzie, Influenza forecasting in human populations: A scoping review, PloS one, 9 (2014), e94130. doi: 10.1371/journal.pone.0094130. Google Scholar

[6]

A. D. Cliff, P. Haggett and J. K. Ord, Spatial Aspects of Influenza Epidemics, Routledge, 1986. Google Scholar

[7]

V. Colizza, A. Barrat, M. Barthelemy, A. -J. Valleron and A. Vespignani, Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions, PLoS Med, 4 (2007), e13.Google Scholar

[8]

S. Cook, C. Conrad, A. L. Fowlkes and M. H. Mohebbi, Assessing google flu trends performance in the united states during the 2009 influenza virus a (h1n1) pandemic PloS one, 6 (2011), e23610. doi: 10.1371/journal.pone.0023610. Google Scholar

[9]

N. Goeyvaerts, L. Willem, K. Van~Kerckhove, Y. Vandendijck, G. Hanquet, P. Beutels and N. Hens, Estimating dynamic transmission model parameters for seasonal influenza by fitting to age and season-specific influenza-like illness incidence Epidemics, 13 (2015), p1. doi: 10.1016/j.epidem.2015.04.002. Google Scholar

[10]

I. HallR. GaniH. Hughes and S. Leach, Real-time epidemic forecasting for pandemic influenza, Epidemiology and Infection, 135 (2007), 372-385. doi: 10.1017/S0950268806007084. Google Scholar

[11]

D. He, J. Dushoff, R. Eftimie and D. J. Earn, Patterns of spread of influenza A in Canada, Proceedings of the Royal Society of London B: Biological Sciences, 280 (2013), 20131174. doi: 10.1098/rspb.2013.1174. Google Scholar

[12]

K. S. HickmannG. FairchildR. PriedhorskyN. GenerousJ. M. HymanA. Deshpande and S. Y. Del Valle, Forecasting the 2013-2014 influenza season using wikipedia, PLoS Comput Biol, 11 (2015), e1004239. doi: 10.1371/journal.pcbi.1004239. Google Scholar

[13]

A. Hyder, D. L. Buckeridge and B. Leung, Predictive validation of an influenza spread model PloS one, 8 (2013), e65459. doi: 10.1371/journal.pone.0065459. Google Scholar

[15]

Y. G. Ivannikov and A. T. Ismagulov, Epidemiologiya Grippa (The Epidemiology of Influenza), Almaty, Kazakhstan, 1983, In Russian.Google Scholar

[16]

Y. Ivannikov and P. Ogarkov, An experience of mathematical computing forecasting of the influenza epidemics for big territory, Journal of Infectology, 4 (2012), 101-106. Google Scholar

[17]

V. N. Leonenko and S. V. Ivanov, Fitting the SEIR model of seasonal influenza outbreak to the incidence data for Russian cities, Russian Journal of Numerical Analysis and Mathematical Modelling, 31 (2016), 267-279. doi: 10.1515/rnam-2016-0026. Google Scholar

[18]

V. N. LeonenkoS. V. Ivanov and Y. K. Novoselova, A computational approach to investigate patterns of acute respiratory illness dynamics in the regions with distinct seasonal climate transitions, Procedia Computer Science, 80 (2016), 2402-2412. doi: 10.1016/j.procs.2016.05.538. Google Scholar

[19]

V. N. LeonenkoY. K. Novoselova and K. M. Ong, Influenza outbreaks forecasting in Russian cities: Is Baroyan-Rvachev approach still applicable?, Procedia Computer Science, 101 (2016), 282-291. doi: 10.1016/j.procs.2016.11.033. Google Scholar

[20]

V. N. LeonenkoN. V. Pertsev and M. Artzrouni, Using high performance algorithms for the hybrid simulation of disease dynamics on CPU and GPU, Procedia Computer Science, 51 (2015), 150-159. doi: 10.1016/j.procs.2015.05.214. Google Scholar

[21]

D. C. Liu and J. Nocedal, On the limited memory bfgs method for large scale optimization, Mathematical programming, 45 (1989), 503-528. doi: 10.1007/BF01589116. Google Scholar

[22]

A. RomanyukhaT. Sannikova and I. Drynov, The origin of acute respiratory epidemics, Herald of the Russian Academy of Sciences, 81 (2011), 31-34. doi: 10.1134/S1019331611010114. Google Scholar

[23]

L. A. Rvachev and I. M. Longini, A mathematical model for the global spread of influenza, Mathematical Biosciences, 75 (1985), 1-22. doi: 10.1016/0025-5564(85)90063-X. Google Scholar

[24]

J. Shaman, V. E. Pitzer, C. Viboud, B. T. Grenfell and M. Lipsitch, Absolute humidity and the seasonal onset of influenza in the continental United States, PLoS Biol, 8 (2010), e1000316.Google Scholar

[25]

J. TameriusM. I. NelsonS. Z. ZhouC. ViboudM. A. Miller and W. J. Alonso, Global influenza seasonality: Reconciling patterns across temperate and tropical regions, Environmental Health Perspectives, 119 (2011), 439-445. doi: 10.1289/ehp.1002383. Google Scholar

[26]

J. TruscottC. FraserS. CauchemezA. MeeyaiW. HinsleyC. A. DonnellyA. Ghani and N. Ferguson, Essential epidemiological mechanisms underpinning the transmission dynamics of seasonal influenza, Journal of The Royal Society Interface, 9 (2011), 304-312. doi: 10.1098/rsif.2011.0309. Google Scholar

[27]

S. P. van NoortR. ÁguasS. Ballesteros and M. G. M. Gomes, The role of weather on the relation between influenza and influenza-like illness, Journal of Theoretical Biology, 298 (2012), 131-137. Google Scholar

[28]

C. ViboudO. N. BjornstadD. L. SmithL. SimonsenM. A. Miller and B. T. Grenfell, Synchrony, waves, and spatial hierarchies in the spread of influenza, Science, 312 (2006), 447-451. doi: 10.1126/science.1125237. Google Scholar

[29]

WHO, Influenza (seasonal). Fact sheet No. 211, March 2014. , [online], http://www.who.int/mediacentre/factsheets/fs211/en/.Google Scholar

[30]

WHO, Surveillance case definitions for ILI and SARI, [online], http://www.who.int/influenza/surveillance_monitoring/ili_sari_surveillance_case_definition/en/.Google Scholar

[31]

R. Yaari, G. Katriel, A. Huppert, J. Axelsen and L. Stone, Modelling seasonal influenza: The role of weather and punctuated antigenic drift, Journal of The Royal Society Interface, 10 (2013), 20130298. doi: 10.1098/rsif.2013.0298. Google Scholar

[32]

W. Yang, B. J. Cowling, E. H. Lau and J. Shaman, Forecasting influenza epidemics in hong kong, PLoS Comput Biol, 11 (2015), e1004383. doi: 10.1371/journal.pcbi.1004383. Google Scholar

Figure 1.  An example of the epidemic curve extraction from the interpolated ARI incidence
Figure 2.  ARI incidence curve showing the discrepancy between different curve allocation algorithms
Figure 3.  The fitting quality of the algorithms in the case of complete outbreak data
Figure 4.  An example of epidemic peak prediction by Baroyan-Rvachev model
Figure 5.  An example of estimating forecast accuracies. The height of the green bars corresponds to the duration of the outbreak before reaching the peak, the markers indicate the day when the accuracy criterion was reached by the particular model. The absent bars correspond to the years without an epidemic outbreak
Figure 6.  The dependence of the percentage of accurate predictions from the number of days before the peak, in overall for the three cities
Figure 7.  The dependence of the percentage of accurate predictions from the number of days before the peak, horizontal stripe
Figure 8.  The dependence of the percentage of accurate predictions from the number of days before the peak, vertical stripe
Figure 9.  The dependence of the percentage of accurate predictions from the number of days before the peak, square
Figure 10.  The comparison of the statistical and modeling forecasting methods
Table 1.  Parameters of the fitting algorithm for the continuous SEIR model
DefinitionDescriptionValueUnit
Epidemiological parameters
$\alpha$Initial ratio of susceptible individuals in the populationEstimated-
$\beta$Intensity of infectionEstimated1/(person$\cdot$day)
$\gamma$Intensity of transition to infective form of the disease0.391/day
$\delta$Intensity of recovery0.1331/day
$I_{0}$Initial ratio of the infected0.0001-
Curve positioning parameters
$k_{inc}$Relative vertical bias of the modeled incidence curve positionEstimated-
$\Delta_s$Absolute horizontal bias of the modeled incidence curve epidemic start position compared to the dataEstimatedday
DefinitionDescriptionValueUnit
Epidemiological parameters
$\alpha$Initial ratio of susceptible individuals in the populationEstimated-
$\beta$Intensity of infectionEstimated1/(person$\cdot$day)
$\gamma$Intensity of transition to infective form of the disease0.391/day
$\delta$Intensity of recovery0.1331/day
$I_{0}$Initial ratio of the infected0.0001-
Curve positioning parameters
$k_{inc}$Relative vertical bias of the modeled incidence curve positionEstimated-
$\Delta_s$Absolute horizontal bias of the modeled incidence curve epidemic start position compared to the dataEstimatedday
Table 2.  Parameters of the fitting algorithm for Baroyan-Rvachev model
DefinitionDescriptionValueUnit
Model parameters
$\alpha$Initial ratio of susceptible individuals in the populationEstimated-
$\beta$Intensity of infectionEstimated-
$I_{0}$Initial ratio of infected in the populationEstimated-
$T$Duration of infectionFixedday
$g_\tau$A fraction of infectious individuals among those who were infected $\tau$ days before the current momentFixed-
$\rho$Population sizeFixedpersons
Curve positioning parameters
$\Delta_p$Absolute horizontal bias of the modeled incidence curve peak position compared to the dataEstimatedday
DefinitionDescriptionValueUnit
Model parameters
$\alpha$Initial ratio of susceptible individuals in the populationEstimated-
$\beta$Intensity of infectionEstimated-
$I_{0}$Initial ratio of infected in the populationEstimated-
$T$Duration of infectionFixedday
$g_\tau$A fraction of infectious individuals among those who were infected $\tau$ days before the current momentFixed-
$\rho$Population sizeFixedpersons
Curve positioning parameters
$\Delta_p$Absolute horizontal bias of the modeled incidence curve peak position compared to the dataEstimatedday
Table 3.  Varied model parameters
DefinitionDescriptionValueVariation type
Continuous SEIR model
$\alpha$Initial ratio of susceptible individuals in the population$[10^{-2}; 1.0]$BFGS optimization
$\beta$Intensity of infection$[10^{-7}; 50.0]$BFGS optimization
$k_{inc}$Relative vertical bias of the modeled incidence curve position$[0.8; 1.0]$BFGS optimization
$\Delta_s$Absolute horizontal bias of the modeled incidence curve epidemic start position compared to the data$5, \dots, 54$Iteration
Baroyan-Rvachev model
$k$The service parameter defining the product of $\alpha$ and $\beta$$[1.02; 1.6]$BFGS optimization
$I_{0}$Initial ratio of infected in the population$[10^{-1}; 50.0]$BFGS optimization
$\Delta_p$*Absolute horizontal bias of the modeled incidence curve peak position compared to the data$-3, \dots, 3$Iteration
$\theta^{(dat)}_{peak}$**Prospected incidence curve peak day$17, \dots, 83$Iteration
  * For complete incidence data ** For incomplete incidence data
DefinitionDescriptionValueVariation type
Continuous SEIR model
$\alpha$Initial ratio of susceptible individuals in the population$[10^{-2}; 1.0]$BFGS optimization
$\beta$Intensity of infection$[10^{-7}; 50.0]$BFGS optimization
$k_{inc}$Relative vertical bias of the modeled incidence curve position$[0.8; 1.0]$BFGS optimization
$\Delta_s$Absolute horizontal bias of the modeled incidence curve epidemic start position compared to the data$5, \dots, 54$Iteration
Baroyan-Rvachev model
$k$The service parameter defining the product of $\alpha$ and $\beta$$[1.02; 1.6]$BFGS optimization
$I_{0}$Initial ratio of infected in the population$[10^{-1}; 50.0]$BFGS optimization
$\Delta_p$*Absolute horizontal bias of the modeled incidence curve peak position compared to the data$-3, \dots, 3$Iteration
$\theta^{(dat)}_{peak}$**Prospected incidence curve peak day$17, \dots, 83$Iteration
  * For complete incidence data ** For incomplete incidence data
[1]

Jacques Demongeot, Mohamad Ghassani, Mustapha Rachdi, Idir Ouassou, Carla Taramasco. Archimedean copula and contagion modeling in epidemiology. Networks & Heterogeneous Media, 2013, 8 (1) : 149-170. doi: 10.3934/nhm.2013.8.149

[2]

Fok Ricky, Lasek Agnieszka, Li Jiye, An Aijun. Modeling daily guest count prediction. Big Data & Information Analytics, 2016, 1 (4) : 299-308. doi: 10.3934/bdia.2016012

[3]

Kasia A. Pawelek, Anne Oeldorf-Hirsch, Libin Rong. Modeling the impact of twitter on influenza epidemics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1337-1356. doi: 10.3934/mbe.2014.11.1337

[4]

Brandy Rapatski, Petra Klepac, Stephen Dueck, Maoxing Liu, Leda Ivic Weiss. Mathematical epidemiology of HIV/AIDS in cuba during the period 1986-2000. Mathematical Biosciences & Engineering, 2006, 3 (3) : 545-556. doi: 10.3934/mbe.2006.3.545

[5]

Avner Friedman, Wenrui Hao. Mathematical modeling of liver fibrosis. Mathematical Biosciences & Engineering, 2017, 14 (1) : 143-164. doi: 10.3934/mbe.2017010

[6]

Yicang Zhou, Yiming Shao, Yuhua Ruan, Jianqing Xu, Zhien Ma, Changlin Mei, Jianhong Wu. Modeling and prediction of HIV in China: transmission rates structured by infection ages. Mathematical Biosciences & Engineering, 2008, 5 (2) : 403-418. doi: 10.3934/mbe.2008.5.403

[7]

Abdessamad Tridane, Yang Kuang. Modeling the interaction of cytotoxic T lymphocytes and influenza virus infected epithelial cells. Mathematical Biosciences & Engineering, 2010, 7 (1) : 171-185. doi: 10.3934/mbe.2010.7.171

[8]

Muhammad Altaf Khan, Muhammad Farhan, Saeed Islam, Ebenezer Bonyah. Modeling the transmission dynamics of avian influenza with saturation and psychological effect. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 455-474. doi: 10.3934/dcdss.2019030

[9]

Gang Bao. Mathematical modeling of nonlinear diffracvtive optics. Conference Publications, 1998, 1998 (Special) : 89-99. doi: 10.3934/proc.1998.1998.89

[10]

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

[11]

Arni S.R. Srinivasa Rao. Modeling the rapid spread of avian influenza (H5N1) in India. Mathematical Biosciences & Engineering, 2008, 5 (3) : 523-537. doi: 10.3934/mbe.2008.5.523

[12]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565

[13]

Karly Jacobsen, Jillian Stupiansky, Sergei S. Pilyugin. Mathematical modeling of citrus groves infected by huanglongbing. Mathematical Biosciences & Engineering, 2013, 10 (3) : 705-728. doi: 10.3934/mbe.2013.10.705

[14]

Bashar Ibrahim. Mathematical analysis and modeling of DNA segregation mechanisms. Mathematical Biosciences & Engineering, 2018, 15 (2) : 429-440. doi: 10.3934/mbe.2018019

[15]

Natalia L. Komarova. Mathematical modeling of cyclic treatments of chronic myeloid leukemia. Mathematical Biosciences & Engineering, 2011, 8 (2) : 289-306. doi: 10.3934/mbe.2011.8.289

[16]

Jeong-Mi Yoon, Volodymyr Hrynkiv, Lisa Morano, Anh Tuan Nguyen, Sara Wilder, Forrest Mitchell. Mathematical modeling of Glassy-winged sharpshooter population. Mathematical Biosciences & Engineering, 2014, 11 (3) : 667-677. doi: 10.3934/mbe.2014.11.667

[17]

Evans K. Afenya. Using Mathematical Modeling as a Resource in Clinical Trials. Mathematical Biosciences & Engineering, 2005, 2 (3) : 421-436. doi: 10.3934/mbe.2005.2.421

[18]

Adélia Sequeira, Rafael F. Santos, Tomáš Bodnár. Blood coagulation dynamics: mathematical modeling and stability results. Mathematical Biosciences & Engineering, 2011, 8 (2) : 425-443. doi: 10.3934/mbe.2011.8.425

[19]

Victor Fabian Morales-Delgado, José Francisco Gómez-Aguilar, Marco Antonio Taneco-Hernández. Mathematical modeling approach to the fractional Bergman's model. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 805-821. doi: 10.3934/dcdss.2020046

[20]

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

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (18)
  • HTML views (140)
  • Cited by (0)

Other articles
by authors

[Back to Top]