JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (10): 1-12.doi: 10.6040/j.issn.1671-9352.0.2023.340

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Peak value and peak time of nonlinear heterogeneous epidemic model

Shengqiang LIU1(),Ningjuan MA2   

  1. 1. School of Mathematical Science, Tiangong University, Tianjin 300387, China
    2. Beijing Bacui Junyuan School, Beijing 101102, China
  • Received:2023-08-04 Online:2023-10-20 Published:2023-10-17

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Abstract:

The study focuses on analyzing the transmission mechanisms of nonlinear heterogeneous epidemic models within a short-time frame. As a result, novel criteria for determining the peak value and peak time of a heterogeneous epidemic model are achieved, and numerical fitting is applied to prove the efficient of the mold.

Key words: epidemic model, short-time scale, social distance, transmission mechanism

CLC Number: 

  • O193

Fig.1

Chart of epidemiological trends for System (1) and (5)"

Fig.2

Chart of epidemiological trends for System (5) and (10)"

Fig.3

Chart of epidemiological trends for System (6) and (11)"

1 BRAUER F , CASTILLO-CHAVEZ C . Mathematical models in population biology and epidemiology[M]. New York: Springer, 2012.
2 BRADY O J , GETHING P W , BHATT S , et al. Refining the global spatial limits of dengue virus transmission by evidence-based consensus[J]. PLoS Neglected Tropical Diseases, 2012, 6 (8): 1760- 1775.
doi: 10.1371/journal.pntd.0001760
3 BHATT S , GETHING P W , BRADY O J , et al. The global distribution and burden of dengue[J]. Nature, 2013, 496 (7446): 504- 507.
doi: 10.1038/nature12060
4 马知恩, 周义仓, 吴建宏. 传染病的建模与动力学[M]. 北京: 高等教育出版社, 2009.
MA Zhieng , ZHOU Yicang , WU Jianhong . Modeling and dynamics of infectious diseases[M]. Beijing: Higher Education Press, 2009.
5 WU Hongjia. Research on data analysis and modeling of sudden major infectious diseases[D]. Tianjin: Tiangong University, 2021.
6 KERMACK W O , MCKENDRICK A G . A contribution to the mathematical theory of epidemics[J]. Proceedings of the Royal Society A(Mathematical, Physical and Engineering Sciences), 1927, 115 (772): 700- 721.
7 KERMACK W O , MCKENDRICK A G . Contributions to the mathematical theory of epidemics. Ⅱ. The problem of endemicity[J]. Proceedings of the Royal Society A(Mathematical, Physical and Engineering Sciences), 1927, 138 (834): 55- 83.
8 ALLEN I J S . Some discrete-time SI, SIR and SIS epidemic models[J]. Mathematical Biosciences, 1994, 124, 83- 105.
doi: 10.1016/0025-5564(94)90025-6
9 ALLEN I J S , Burgin A M . Comparison of deterministic and stochastic SIS and SIR models in discrete time[J]. Mathematical Biosciences, 2000, 163, 1- 33.
doi: 10.1016/S0025-5564(99)00047-4
10 BRAUER F . Some simple nosocomial disease transmission models[J]. Bulletin of Mathematical Biology, 2015, 77 (3): 460- 469.
doi: 10.1007/s11538-015-0061-0
11 FENG Zhilan . Final and peak epidemic sizes for SEIR models with quarantine and isolation[J]. Mathematical Biosciences and Engineering, 2007, 4 (4): 675- 686.
doi: 10.3934/mbe.2007.4.675
12 ZHAO Shi , MUSA S S , LIN Qianying , et al. Estimating the unreported number of novel coronavirus (2019-nCoV) cases in China in the first half of January 2020: a data-driven modelling analysis of the early outbreak[J]. Journal of Clinical Medicine, 2020, 9 (2): 388.
doi: 10.3390/jcm9020388
13 LI Qun , GUAN Xuhua , WU Peng , et al. Early transmissiodynamics in Wuhan, China, of novel coronavirus—infected pneumonia[J]. New England Journal of Medicine, 2020, 382 (13): 1199- 1207.
doi: 10.1056/NEJMoa2001316
14 WU J T , KATHY L , LEUNG G M . Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study[J]. Lancet(London, England), 2020, 395 (10225): 689- 697.
15 TANG Biao , WANG Xian , LI Qiqn , et al. Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions[J]. Journal of Clinical Medicine, 2020, 9 (2): 462- 463.
doi: 10.3390/jcm9020462
16 SHANAFELT D W , JONES G , LIMA M , et al. Forecasting the 2001 foot-and-mouth disease epidemic in the UK[J]. Ecohealth, 2018, 15 (2): 338- 347.
doi: 10.1007/s10393-017-1293-2
17 WONG N S , LEE S S , MITCHELL KM , et al. Impact of pre-event testing and quarantine on reducing the risk of COVID-19 epidemic rebound: a modelling study[J]. BMC Infectious Diseases, 2022, 22 (1): 83.
doi: 10.1186/s12879-021-06963-2
18 SUN Ruoyan , SHI Junping . Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates[J]. Applied Mathematics and Computation, 2011, 218 (2): 280- 286.
doi: 10.1016/j.amc.2011.05.056
19 MUROYA Y , ENATSU Y , LI H X . A note on the global stability of an SEIR epidemic model with constant latency time and infectious period[J]. Discrete and Continuous Dynamical Systems(Series B), 2012, 18 (1): 173- 183.
20 CHEN Xiaodan D , CUI Renhao . Global stability in a diffusive cholera epidemic model with nonlinear incidence[J]. Applied Mathematics Letters, 2020, 111, 106596.
21 WANG Xinxin , LIU Shengqiang , WANG Lin , et al. An epidemic patchy model with entry-exit screening[J]. Bulletin of Mathematical Biology, 2015, 77 (7): 1237- 1255.
doi: 10.1007/s11538-015-0084-6
22 FOY H M , COONEY M K , ALLAN I . Longitudinal studies of types A and B influenza among Seattle school children and families[J]. Journal of Infectious Diseases, 1976, 134 (4): 362- 369.
doi: 10.1093/infdis/134.4.362
23 LOEB M , RUSSELL M L , MOSS L , et al. Effect of influenza vaccination of children on infection rates in Hutterite communities: a randomized trial[J]. Journal of the American Medical Association, 2010, 303 (10): 943- 950.
doi: 10.1001/jama.2010.250
24 MUROYA Y , ENATSUB Y , KUNIYA T . Global stability for a multi-group SIRS epidemic model with varying population sizes[J]. Nonlinear Analysis(Real World Applications), 2013, 14 (3): 1693- 1704.
doi: 10.1016/j.nonrwa.2012.11.005
25 WANG Jinliang , WANG Jing , KUNIYA T . Analysis of an age-structured multi-group heroin epidemic model[J]. Applied Mathematics and Computation, 2019, 347, 78- 100.
doi: 10.1016/j.amc.2018.11.012
26 WANG Xia , WU Huilin , TANG Sanyi . Assessing age-specific vaccination strategies and post-vaccination reopening policies for COVID-19 Control Using SEIR modeling approach[J]. Bulletin of Mathematical Biology, 2022, 84 (10): 108- 120.
doi: 10.1007/s11538-022-01064-w
27 BRAUER F . Early estimates of epidemic final sizes[J]. Journal of Biological Dynamics, 2018, 13 (SI): 23- 30.
28 BRAUER F . The final size of a serious epidemic[J]. Bulletin of Mathematical Biology, 2018, 81 (3): 869- 877.
29 CUI Jingan , ZHANG Yannan , FENG Zhilan . Influence of non-homogeneous mixing on final epidemic size in a meta-population model[J]. Journal of Biological Dynamics, 2018, 13 (SI): 31- 46.
30 ARINO J , BRAUER F , VAN DEN DRIESSCHE P , et al. Simple models for containment of a pandemic[J]. Journal of the Royal Society Interface, 2006, 3 (8): 453- 457.
doi: 10.1098/rsif.2006.0112
31 ARINO J , BRAUER F , VAN DEN DRIESSCHE P , et al. A final size relation for epidemic models[J]. Mathematical Biosciences and Engineering, 2007, 4 (2): 159- 175.
doi: 10.3934/mbe.2007.4.159
32 BRAUER F . The Kermack-McKendrick epidemic model revisited[J]. Mathematical Biosciences, 2005, 198 (2): 119- 131.
doi: 10.1016/j.mbs.2005.07.006
33 MA J , EARN D J D . Generality of the final size formula for an epidemic of a newly invading infectious disease[J]. Bulletin of Mathematical Biology, 2006, 68 (3): 679- 702.
doi: 10.1007/s11538-005-9047-7
34 LI Jianquan , LI Yiqun , YANG Yali , et al. Epidemic characteristics of two classic models and the dependence on the initial conditions[J]. Mathematical Biosciences and Engineering, 2016, 13 (5): 999- 1010.
doi: 10.3934/mbe.2016027
35 LI Jianquan , LOU Yajian . Characteristics of an epidemic outbreak with a large initial infection size[J]. Journal of Biological Dynamics, 2016, 10 (1): 366- 378.
doi: 10.1080/17513758.2016.1205223
36 CHEN Yan , SONG Haitao , LIU Shengqiang . Evaluations of COVID-19 epidemic models with multiple susceptible compartments using exponential and non-exponential distribution for disease stag[J]. Infectious Disease Modelling, 2022, 7, 795- 810.
doi: 10.1016/j.idm.2022.11.004
37 DOLBEAULT J , TURINICI G . Heterogeneous social interactions and the COVID-19 lockdown outcome in a multi-group SEIR model[J]. Mathematical Modelling of Natural Phenomena, 2020, 15, 36.
doi: 10.1051/mmnp/2020025
38 LIU Shengqiang , WANG Shengkai , WANG Lin . Global dynamics of delay epidemic models with nonlinear incidence rate and relapse[J]. Nonlinear Analysis(Real World Applications), 2011, 12, 119- 127.
doi: 10.1016/j.nonrwa.2010.06.001
39 RUAN Shigui , WANG Wendi . Dynamical behavior of an epidemic model with nonlinear incidence rate[J]. Journal of Differential Equations, 2003, 188 (1): 135- 163.
doi: 10.1016/S0022-0396(02)00089-X
40 HETHCOTE H W , DRIESSCHE P . Some epidemiological models with nonlinear incidence[J]. Journal of Mathematical Biology, 1991, 29 (3): 271- 287.
doi: 10.1007/BF00160539
41 ANDERSON R M , MAY R M . Population biology of infectious diseases Ⅰ[J]. Nature, 1979, 180, 361- 367.
42 ANDERSON R M , MAY R M . Infectious diseases of humans, dynamics and control[M]. Oxford: Oxford University Press, 1992.
43 COOKE K L , VAN DEN DRIESSCHE P . Analysis of an SEIRS epidemic model with two delays[J]. Journal of Mathematical Biology, 1996, 35 (2): 240- 260.
doi: 10.1007/s002850050051
44 WANG Wendi . Global behavior of an SEIRS epidemic model with time delays[J]. Applied Mathematics Letters, 2002, 15 (4): 423- 428.
doi: 10.1016/S0893-9659(01)00153-7
45 HUI J , CHEN L . Impulsive vaccination of SIR epidemic models with nonlinear incidence rates[J]. Discrete and Continuous Dynamical Systems(Series B), 2004, 3 (4): 595- 605.
46 MAY R M , ANDERSON R M . Regulation and stability of host-parasite population interactions Ⅱ: destabilizing process[J]. Journal of Animal Ecology, 1978, 47, 219- 267.
doi: 10.2307/3933
47 DIEKMANN O , HEESTERBEEK J A P , METZ J A J . On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations[J]. Journal of Mathematical Biology, 1990, 28, 365- 382.
48 BLACKWOOD J C , CHILDS L M . An introduction to compartmental modeling for the budding infectious disease modeler[J]. Letters in Biomathematics, 2018, 5, 195- 221.
doi: 10.30707/LiB5.1Blackwood
49 VAN DEN DRIESSCHE P , WATMOUGH J . Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J]. Mathematical Biosciences, 2002, 180, 29- 48.
doi: 10.1016/S0025-5564(02)00108-6
50 DIEKMANN O , HEESTERBEEK J A P , ROBERTS M G . The construction of next-generation matrices for compartmental epidemic models[J]. Journal of the Royal Society Interface, 2009, 7 (47): 873- 885.
51 FENG Zhilan . Final and peak epidemic sizes for SEIR models with quarantine and isolation[J]. Mathematical Biosciences and Engineering, 2007, 4 (4): 675- 686.
doi: 10.3934/mbe.2007.4.675
52 BLACKWOOD J C , CHILDS L M . An introduction to compartmental modeling for the budding infectious disease modeler[J]. Letters in Biomathematics, 2018, 5 (1): 195- 221.
doi: 10.30707/LiB5.1Blackwood
53 BACAER N . Un modèle mathématique des débuts de l'épidémie de coronavirus en France[J]. Mathematical Modelling of Natural Phenomena, 2020, 15, 29- 30.
doi: 10.1051/mmnp/2020015
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