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December  2018, 15(6): 1425-1434. doi: 10.3934/mbe.2018065

Impact of behavioral change on the epidemic characteristics of an epidemic model without vital dynamics

School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an, Shaanxi 710021, China

* Corresponding author: Jianquan Li

Received  January 14, 2018 Accepted  June 11, 2018 Published  September 2018

Fund Project: The authors were supported by National Natural Science Foundation of China (11371369, 11771259).

The epidemic characteristics of an epidemic model with behavioral change in [V. Capasso, G. Serio, A generalizaition of the Kermack-McKendrick deterministic epidemic model, Math. Bios., 42 (1978), 43-61] are investigated, including the epidemic size, peak and turning point. The conditions on the appearance of the peak state and turning point are represented clearly, and the expressions determining the corresponding time for the peak state and turning point are described explicitly. Moreover, the impact of behavioral change on the characteristics is discussed.

Citation: Jianquan Li, Xiaoqin Wang, Xiaolin Lin. Impact of behavioral change on the epidemic characteristics of an epidemic model without vital dynamics. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1425-1434. doi: 10.3934/mbe.2018065
References:
[1]

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

[2]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, 2nd edn. Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[3]

V. Capasso and G. Serio, A generalizaition of the Kermack-McKendrick deterministic epidemic model, Math. Bios., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[4]

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

[5]

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

[6]

J. LiY. Li and Y. Yang, Epidemic characteristics of two classic models and the dependence on the initial conditions, Math. Bios. Eng., 13 (2016), 999-1010.  doi: 10.3934/mbe.2016027.  Google Scholar

[7]

J. Li and Y. Lou, Characteristics of an epidemic outbreak with a large initial infection size, J. Biol. Dyn., 10 (2016), 366-378.  doi: 10.1080/17513758.2016.1205223.  Google Scholar

[8]

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

[9]

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

[10]

J. C. Miller, A note on the derivation of epidemic final sizes, Bull. Math. Biol., 74 (2012), 2125-2141.  doi: 10.1007/s11538-012-9749-6.  Google Scholar

[11]

F. ZhangJ. Li and J. Li, Epidemic characteristics of two classic SIS models with disease-induced death, J. Theoret. Biol., 424 (2017), 73-83.  doi: 10.1016/j.jtbi.2017.04.029.  Google Scholar

show all references

References:
[1]

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

[2]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, 2nd edn. Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[3]

V. Capasso and G. Serio, A generalizaition of the Kermack-McKendrick deterministic epidemic model, Math. Bios., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[4]

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

[5]

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

[6]

J. LiY. Li and Y. Yang, Epidemic characteristics of two classic models and the dependence on the initial conditions, Math. Bios. Eng., 13 (2016), 999-1010.  doi: 10.3934/mbe.2016027.  Google Scholar

[7]

J. Li and Y. Lou, Characteristics of an epidemic outbreak with a large initial infection size, J. Biol. Dyn., 10 (2016), 366-378.  doi: 10.1080/17513758.2016.1205223.  Google Scholar

[8]

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

[9]

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

[10]

J. C. Miller, A note on the derivation of epidemic final sizes, Bull. Math. Biol., 74 (2012), 2125-2141.  doi: 10.1007/s11538-012-9749-6.  Google Scholar

[11]

F. ZhangJ. Li and J. Li, Epidemic characteristics of two classic SIS models with disease-induced death, J. Theoret. Biol., 424 (2017), 73-83.  doi: 10.1016/j.jtbi.2017.04.029.  Google Scholar

Figure 1.  The case that both the peak and the turning point can appear
Figure 2.  The case that there is the peak and no turning point
Figure 3.  The case that both the peak and the turning point can not appear
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