American Institute of Mathematical Sciences

Advanced
2016, 13(5): 999-1010. doi: 10.3934/mbe.2016027

Epidemic characteristics of two classic models and the dependence on the initial conditions

Jianquan Li 1, , Yiqun Li 1, and  Yali Yang 2,

1. 

Science College, Air Force Engineering University, Xi'an 710051, China
2. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062

Received  December 2015 Revised  May 2016 Published  July 2016

The epidemic characteristics, including the epidemic final size, peak, and turning point, of two classical SIR models with disease-induced death are investigated when a small initial value of the infective population is released. The models have mass-action (i.e. bilinear), or density dependent (i.e. standard) incidence, respectively. For the two models, the conditions that determining whether the related epidemic characteristics of an epidemic outbreak appear are explicitly determine by rigorous mathematical analysis. The dependence of the epidemic final size on the initial values of the infective class is demonstrated. The peak, turning point (if it exists) and the corresponding time are found. The obtained results suggest that their basic reproduction numbers are one factor determining the epidemic characteristics, but not the only one. The characteristics of the two models depend on the initial values and proportions of various compartments as well. At last, the similarities and differences of the epidemic characteristics between the two models are discussed.
Keywords: Epidemic models, peak, turning point., epidemic final size.
Mathematics Subject Classification: Primary: 92D3.
Citation: Jianquan Li, Yiqun Li, Yali Yang. Epidemic characteristics of two classic models and the dependence on the initial conditions. Mathematical Biosciences & Engineering, 2016, 13 (5) : 999-1010. doi: 10.3934/mbe.2016027
References:
[1]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, Simple models for containment of a pandemic, J. R. Soc. Interface, 3 (2006), 453-457. doi: 10.1098/rsif.2006.0112.  Google Scholar

[2]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, A final size relation for epidemic models, Math. Bios. Eng., 4 (2007), 159-175. doi: 10.3934/mbe.2007.4.159.  Google Scholar

[3]

N. Bacaër and M. G. M. Gomes, On the final size of epidemics with seasonality, Bull. Math. Biol., 71 (2009), 1954-1966. doi: 10.1007/s11538-009-9433-7.  Google Scholar

[4]

F. Brauer, Some simple epidemic models, Math. Bios. Eng., 3 (2006), 1-15. doi: 10.3934/mbe.2006.3.1.  Google Scholar

[5]

F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Bios., 198 (2005), 119-131. doi: 10.1016/j.mbs.2005.07.006.  Google Scholar

[6]

F. Brauer, Some simple nosocomial disease transmission models, Bull. Math. Biol., 77 (2015), 460-469. doi: 10.1007/s11538-015-0061-0.  Google Scholar

[7]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[8]

D. L. Chao and D. T. Dimitrov, Seasonality and the effectiveness of mass vaccination, Math. Bios. Eng., 13 (2016), 249-259. doi: 10.3934/mbe.2015001.  Google Scholar

[9]

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), 365-382. doi: 10.1007/BF00178324.  Google Scholar

[10]

O. Diekmann, J. A. J. Metz and J. A. P. Heesterbeek, The legacy of Kermack and McKendrick, In: Epidemic Models: Their Structure and Relation to Data, Their Structure and Relation to Data, Cambridge University Press, 1995, 95-115. Google Scholar

[11]

A. Ed-Darraz and M. Khaladi, On the final size of epidemics in random environment, Math. Bios., 266 (2015), 10-14. doi: 10.1016/j.mbs.2015.05.004.  Google Scholar

[12]

Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Bios. Eng., 4 (2007), 675-686. doi: 10.3934/mbe.2007.4.675.  Google Scholar

[13]

T. House, J. V. Ross and D. Sirl, How big is an outbreak likely to be? Methods for epidemic final-size calculation, Proc. R. Soc. A, 469 (2013), 20120436, 22 pp. doi: 10.1098/rspa.2012.0436.  Google Scholar

[14]

Y. H. Hsieh, Pandemic influenza A (H1N1) during winter influenza season in the southern hemisphere, Infl. Other Resp. Vir. 4 (2010), 187-197. doi: 10.1111/j.1750-2659.2010.00147.x.  Google Scholar

[15]

Y. H. Hsieh, Richards model: a simple procedure for real-time prediction of outbreak severity. In: Z. Ma, J. Wu and Y. Zhou editors, Modeling and Dynamics of Infectious Diseases, Series in Contemporary Applied Mathematics (CAM), Higher Education Press, Beijing, 2009, 216-236.  Google Scholar

[16]

Y. H. Hsieh, H. de Arazoza and R. Lounes, Temporal trends and regional variability of 2001-2002 multiwave DENV-3 epidemic in Havana City: did Hurricane Michelle contribute to its severity? Trop. Med. Int. Health., 18 (2013), 830-838. doi: 10.1111/tmi.12105.  Google Scholar

[17]

Y. H. Hsieh and C. W. S. Chen, Turning points, reproduction number, and impact of climatological events for multi-wave dengue outbreaks, Trop. Med. Int. Heal., 14 (2009), 628-638. doi: 10.1111/j.1365-3156.2009.02277.x.  Google Scholar

[18]

Y. H. Hsieh and Y. S. Cheng, Real-time forecast of multiphase outbreak, Emerg. Infect. Dis., 12 (2006), 122-127. doi: 10.3201/eid1201.050396.  Google Scholar

[19]

Y. H. Hsieh, D. N. Fisman and J. Wu, On epidemic modeling in real time: An application to the 2009 novel A (H1N1) influenza outbreak in Canada, BMC Research Notes, 3 (2010), p283. doi: 10.1186/1756-0500-3-283.  Google Scholar

[20]

Y. H. Hsieh, J. Y. Lee and H. L. Chang, SARS epidemiology modeling, Emerg. Infect. Dis., 10 (2004), 1165-1167. doi: 10.3201/eid1006.031023.  Google Scholar

[21]

Y. H. Hsieh and S. Ma, Intervention measures, turning point, and reproduction number for dengue, Singapore, Am. J. Trop. Med. Hyg., 80 (2005), 66-71. Google Scholar

[22]

Y. H. Hsieh, J. Wu, J. Fang, Y. Yang and J. Lou, Quantification of bird-to-bird and bird-to-human infections during 2013 novel H7N9 avian influenza outbreak in China, PLoS ONE, 9 (2014), e111834. doi: 10.1371/journal.pone.0111834.  Google Scholar

[23]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348. doi: 10.1007/s00285-011-0463-z.  Google Scholar

[24]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. Google Scholar

[25]

J. Ma and D. J. D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol., 68 (2006), 679-702. doi: 10.1007/s11538-005-9047-7.  Google Scholar

[26]

Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, Singapore, 2009. doi: 10.1142/9789812797506.  Google Scholar

[27]

J. C. Miller, Epidemics on networks with large initial conditions or changing structure, PLoS ONE, 9 (2014), e101421. doi: 10.1371/journal.pone.0101421.  Google Scholar

[28]

F. J. Richards, A flexible growth function for empirical use, J. Exp. Bot., 10 (1959), 290-301. doi: 10.1093/jxb/10.2.290.  Google Scholar

[29]

J. V. Ross, A note on density-dependence in population models, Ecol. Model., 220 (2009), 3472-3474. doi: 10.1016/j.ecolmodel.2009.08.024.  Google Scholar

[30]

I. Sazonov, M. Kelbert and M. B. Gravenor, A two-stage model for the SIR outbreak: Accounting for the discrete and stochastic nature of the epidemic at the initial contamination stage, Math. Biosci., 234 (2011), 108-117. doi: 10.1016/j.mbs.2011.09.002.  Google Scholar

[31]

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

[32]

W. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942.  Google Scholar

[33]

B. G. Wang and X. Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models. J. Dyn. Differ. Equ., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7.  Google Scholar

[34]

W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.  Google Scholar

[35]

X. S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics, J. Theo. Biol., 313 (2012), 12-19. doi: 10.1016/j.jtbi.2012.07.024.  Google Scholar

[36]

X. S. Wang and L. Zhong, Ebola outbreak in West Africa: Real-time estimation and multiple-wave prediction, Math. Bios. Eng., 12 (2015), 1055-1063. doi: 10.3934/mbe.2015.12.1055.  Google Scholar

[37]

Y. Xiao, F. Brauer and S. M. Moghadas, Can treatment increase the epidemic size? J. Math. Biol., 72 (2016), 343-361. doi: 10.1007/s00285-015-0887-y.  Google Scholar

[38]

X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Diff. Equat., (2015), 1-16. doi: 10.1007/s10884-015-9425-2.  Google Scholar

show all references

References:
[1]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, Simple models for containment of a pandemic, J. R. Soc. Interface, 3 (2006), 453-457. doi: 10.1098/rsif.2006.0112.  Google Scholar

[2]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, A final size relation for epidemic models, Math. Bios. Eng., 4 (2007), 159-175. doi: 10.3934/mbe.2007.4.159.  Google Scholar

[3]

N. Bacaër and M. G. M. Gomes, On the final size of epidemics with seasonality, Bull. Math. Biol., 71 (2009), 1954-1966. doi: 10.1007/s11538-009-9433-7.  Google Scholar

[4]

F. Brauer, Some simple epidemic models, Math. Bios. Eng., 3 (2006), 1-15. doi: 10.3934/mbe.2006.3.1.  Google Scholar

[5]

F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Bios., 198 (2005), 119-131. doi: 10.1016/j.mbs.2005.07.006.  Google Scholar

[6]

F. Brauer, Some simple nosocomial disease transmission models, Bull. Math. Biol., 77 (2015), 460-469. doi: 10.1007/s11538-015-0061-0.  Google Scholar

[7]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[8]

D. L. Chao and D. T. Dimitrov, Seasonality and the effectiveness of mass vaccination, Math. Bios. Eng., 13 (2016), 249-259. doi: 10.3934/mbe.2015001.  Google Scholar

[9]

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), 365-382. doi: 10.1007/BF00178324.  Google Scholar

[10]

O. Diekmann, J. A. J. Metz and J. A. P. Heesterbeek, The legacy of Kermack and McKendrick, In: Epidemic Models: Their Structure and Relation to Data, Their Structure and Relation to Data, Cambridge University Press, 1995, 95-115. Google Scholar

[11]

A. Ed-Darraz and M. Khaladi, On the final size of epidemics in random environment, Math. Bios., 266 (2015), 10-14. doi: 10.1016/j.mbs.2015.05.004.  Google Scholar

[12]

Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Bios. Eng., 4 (2007), 675-686. doi: 10.3934/mbe.2007.4.675.  Google Scholar

[13]

T. House, J. V. Ross and D. Sirl, How big is an outbreak likely to be? Methods for epidemic final-size calculation, Proc. R. Soc. A, 469 (2013), 20120436, 22 pp. doi: 10.1098/rspa.2012.0436.  Google Scholar

[14]

Y. H. Hsieh, Pandemic influenza A (H1N1) during winter influenza season in the southern hemisphere, Infl. Other Resp. Vir. 4 (2010), 187-197. doi: 10.1111/j.1750-2659.2010.00147.x.  Google Scholar

[15]

Y. H. Hsieh, Richards model: a simple procedure for real-time prediction of outbreak severity. In: Z. Ma, J. Wu and Y. Zhou editors, Modeling and Dynamics of Infectious Diseases, Series in Contemporary Applied Mathematics (CAM), Higher Education Press, Beijing, 2009, 216-236.  Google Scholar

[16]

Y. H. Hsieh, H. de Arazoza and R. Lounes, Temporal trends and regional variability of 2001-2002 multiwave DENV-3 epidemic in Havana City: did Hurricane Michelle contribute to its severity? Trop. Med. Int. Health., 18 (2013), 830-838. doi: 10.1111/tmi.12105.  Google Scholar

[17]

Y. H. Hsieh and C. W. S. Chen, Turning points, reproduction number, and impact of climatological events for multi-wave dengue outbreaks, Trop. Med. Int. Heal., 14 (2009), 628-638. doi: 10.1111/j.1365-3156.2009.02277.x.  Google Scholar

[18]

Y. H. Hsieh and Y. S. Cheng, Real-time forecast of multiphase outbreak, Emerg. Infect. Dis., 12 (2006), 122-127. doi: 10.3201/eid1201.050396.  Google Scholar

[19]

Y. H. Hsieh, D. N. Fisman and J. Wu, On epidemic modeling in real time: An application to the 2009 novel A (H1N1) influenza outbreak in Canada, BMC Research Notes, 3 (2010), p283. doi: 10.1186/1756-0500-3-283.  Google Scholar

[20]

Y. H. Hsieh, J. Y. Lee and H. L. Chang, SARS epidemiology modeling, Emerg. Infect. Dis., 10 (2004), 1165-1167. doi: 10.3201/eid1006.031023.  Google Scholar

[21]

Y. H. Hsieh and S. Ma, Intervention measures, turning point, and reproduction number for dengue, Singapore, Am. J. Trop. Med. Hyg., 80 (2005), 66-71. Google Scholar

[22]

Y. H. Hsieh, J. Wu, J. Fang, Y. Yang and J. Lou, Quantification of bird-to-bird and bird-to-human infections during 2013 novel H7N9 avian influenza outbreak in China, PLoS ONE, 9 (2014), e111834. doi: 10.1371/journal.pone.0111834.  Google Scholar

[23]

H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348. doi: 10.1007/s00285-011-0463-z.  Google Scholar

[24]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. Google Scholar

[25]

J. Ma and D. J. D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol., 68 (2006), 679-702. doi: 10.1007/s11538-005-9047-7.  Google Scholar

[26]

Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, Singapore, 2009. doi: 10.1142/9789812797506.  Google Scholar

[27]

J. C. Miller, Epidemics on networks with large initial conditions or changing structure, PLoS ONE, 9 (2014), e101421. doi: 10.1371/journal.pone.0101421.  Google Scholar

[28]

F. J. Richards, A flexible growth function for empirical use, J. Exp. Bot., 10 (1959), 290-301. doi: 10.1093/jxb/10.2.290.  Google Scholar

[29]

J. V. Ross, A note on density-dependence in population models, Ecol. Model., 220 (2009), 3472-3474. doi: 10.1016/j.ecolmodel.2009.08.024.  Google Scholar

[30]

I. Sazonov, M. Kelbert and M. B. Gravenor, A two-stage model for the SIR outbreak: Accounting for the discrete and stochastic nature of the epidemic at the initial contamination stage, Math. Biosci., 234 (2011), 108-117. doi: 10.1016/j.mbs.2011.09.002.  Google Scholar

[31]

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

[32]

W. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942.  Google Scholar

[33]

B. G. Wang and X. Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models. J. Dyn. Differ. Equ., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7.  Google Scholar

[34]

W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.  Google Scholar

[35]

X. S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics, J. Theo. Biol., 313 (2012), 12-19. doi: 10.1016/j.jtbi.2012.07.024.  Google Scholar

[36]

X. S. Wang and L. Zhong, Ebola outbreak in West Africa: Real-time estimation and multiple-wave prediction, Math. Bios. Eng., 12 (2015), 1055-1063. doi: 10.3934/mbe.2015.12.1055.  Google Scholar

[37]

Y. Xiao, F. Brauer and S. M. Moghadas, Can treatment increase the epidemic size? J. Math. Biol., 72 (2016), 343-361. doi: 10.1007/s00285-015-0887-y.  Google Scholar

[38]

X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Diff. Equat., (2015), 1-16. doi: 10.1007/s10884-015-9425-2.  Google Scholar

[1]

Julien Arino, Fred Brauer, P. van den Driessche, James Watmough, Jianhong Wu. A final size relation for epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (2) : 159-175. doi: 10.3934/mbe.2007.4.159
[2]

Z. Feng. Final and peak epidemic sizes for SEIR models with quarantine and isolation. Mathematical Biosciences & Engineering, 2007, 4 (4) : 675-686. doi: 10.3934/mbe.2007.4.675
[3]

Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 635-642. doi: 10.3934/dcdsb.2004.4.635
[4]

Luis F. Gordillo, Stephen A. Marion, Priscilla E. Greenwood. The effect of patterns of infectiousness on epidemic size. Mathematical Biosciences & Engineering, 2008, 5 (3) : 429-435. doi: 10.3934/mbe.2008.5.429
[5]

E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229
[6]

Fred Brauer. Some simple epidemic models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 1-15. doi: 10.3934/mbe.2006.3.1
[7]

Fred Brauer, Zhilan Feng, Carlos Castillo-Chávez. Discrete epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (1) : 1-15. doi: 10.3934/mbe.2010.7.1
[8]

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
[9]

James M. Hyman, Jia Li. Differential susceptibility and infectivity epidemic models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 89-100. doi: 10.3934/mbe.2006.3.89
[10]

Qingming Gou, Wendi Wang. Global stability of two epidemic models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 333-345. doi: 10.3934/dcdsb.2007.8.333
[11]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445
[12]

Wendi Wang. Epidemic models with nonlinear infection forces. Mathematical Biosciences & Engineering, 2006, 3 (1) : 267-279. doi: 10.3934/mbe.2006.3.267
[13]

Zhen Jin, Guiquan Sun, Huaiping Zhu. Epidemic models for complex networks with demographics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1295-1317. doi: 10.3934/mbe.2014.11.1295
[14]

Yanan Zhao, Daqing Jiang, Xuerong Mao, Alison Gray. The threshold of a stochastic SIRS epidemic model in a population with varying size. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1277-1295. doi: 10.3934/dcdsb.2015.20.1277
[15]

W. E. Fitzgibbon, M.E. Parrott, Glenn Webb. Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 35-57. doi: 10.3934/dcds.1995.1.35
[16]

Fang Li, Nung Kwan Yip. Long time behavior of some epidemic models. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 867-881. doi: 10.3934/dcdsb.2011.16.867
[17]

Dirk Stiefs, Ezio Venturino, Ulrike Feudel. Evidence of chaos in eco-epidemic models. Mathematical Biosciences & Engineering, 2009, 6 (4) : 855-871. doi: 10.3934/mbe.2009.6.855
[18]

Junling Ma, Zhien Ma. Epidemic threshold conditions for seasonally forced SEIR models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 161-172. doi: 10.3934/mbe.2006.3.161
[19]

Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability for a class of discrete SIR epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (2) : 347-361. doi: 10.3934/mbe.2010.7.347
[20]

Eunha Shim. A note on epidemic models with infective immigrants and vaccination. Mathematical Biosciences & Engineering, 2006, 3 (3) : 557-566. doi: 10.3934/mbe.2006.3.557

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences