# American Institute of Mathematical Sciences

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

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

 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.
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. [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. [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. [4] F. Brauer, Some simple epidemic models, Math. Bios. Eng., 3 (2006), 1-15. doi: 10.3934/mbe.2006.3.1. [5] F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Bios., 198 (2005), 119-131. doi: 10.1016/j.mbs.2005.07.006. [6] F. Brauer, Some simple nosocomial disease transmission models, Bull. Math. Biol., 77 (2015), 460-469. doi: 10.1007/s11538-015-0061-0. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, Singapore, 2009. doi: 10.1142/9789812797506. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [4] F. Brauer, Some simple epidemic models, Math. Bios. Eng., 3 (2006), 1-15. doi: 10.3934/mbe.2006.3.1. [5] F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Bios., 198 (2005), 119-131. doi: 10.1016/j.mbs.2005.07.006. [6] F. Brauer, Some simple nosocomial disease transmission models, Bull. Math. Biol., 77 (2015), 460-469. doi: 10.1007/s11538-015-0061-0. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, Singapore, 2009. doi: 10.1142/9789812797506. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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