doi: 10.3934/cpaa.2021155
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Transmission dynamics and high infectiousness of Coronavirus Disease 2019

1. 

Institute of NBC Defense of PLA, Beijing 102205, China

2. 

Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

4. 

Ministry of Education Key Laboratory of Environment and Health, State Key Laboratory of Environmental Health (Incubating), School of Public Health, Tongji Medical College, Huazhong University of Science and Technology, Wuhan 430063, China

5. 

Institute of Biotechnology, Academy of Military Medical Sciences, Beijing 100071, China

6. 

XiangYa School of Public Health, Central South University, Changsha, Hunan 410078, China

* Corresponding author

Contributed equally to this article

Received  February 2021 Revised  July 2021 Early access September 2021

Coronavirus disease 2019 (COVID-19) has rapidly spread around the world since the early 2020. Recently, a second wave of COVID-19 has resurged in many countries. The transmission dynamics and infectiousness of the COVID-19 pandemic remain unclear, and developing strategies to mitigate the severity of the pandemic is a top priority for global public health. According to the infection mechanism of COVID-19, a novel susceptible-asymptomatic-symptomatic-recovered (SASR) model with control variables in a patchy environment was proposed not only to consider the key characteristics of asymptomatic infection and the effects of seasonal variation but also to incorporate different control measures for multiple transmission routes. The basic reproduction number $ R_{0} $ was established to describe the spreading behavior in the natural state over a long time horizon, and the natural reproduction number $ R_{n} $, which describes the development trend of the disease during a short time in the future, was defined according to the actual propagation characteristics. In addition, the effective reproduction number $ R_{e} $ considering the control strategies was proposed to evaluate the impact of non-pharmaceutical interventions. The results of numerical simulations for COVID-19 cases in Wuhan, China, based on the SASR model indicate that $ R_{0} $ was 3.58, $ R_{n} $ ranged from 2.37 to 4.91, and $ R_{e} $ decreased gradually from 4.83 on December 8, 2019 to 0.31 on March 8, 2020, reaching 1.40 on January 23, 2020, when the lockdown was lifted in Wuhan. We further concluded that the total number of infections, including asymptomatic infections, was approximately 301, 804 as of March 8, 2020, in Wuhan, China. In particular, this article proposes a dynamic method to distinguish the impact of natural factors and human interventions on the development of the pandemic, and provides a theoretical basis for fighting the global COVID-19 pandemic.

Citation: Shunxiang Huang, Lin Wu, Jing Li, Ming-Zhen Xin, Yingying Wang, Xingjie Hao, Zhongyi Wang, Qihong Deng, Bin-Guo Wang. Transmission dynamics and high infectiousness of Coronavirus Disease 2019. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021155
References:
[1]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.  Google Scholar

[2]

Y. BaiL. Yao and T. Wei, Presumed asymptomatic carrier transmission of COVID-19, JAMA, 323 (2020), 790-808.  doi: 10.1001/jama.2020.2565.  Google Scholar

[3]

J. F. ChanS. Yuan and K. H. Kok, A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-person transmission: a study of a family cluster, The Lancet, 395 (2020), 514-523.  doi: 10.1016/S0140-6736(20)30154-9.  Google Scholar

[4]

C. Corduneanu, Almost Periodic Functions, New York, 1994. Google Scholar

[5]

COVID-19 prevention and control expert group of Chinese preventive medicine association. The latest understanding of epidemiological characteristics in COVID-19, Chinese J. Viral Diseases, 10 (2003), 86–92(Chinese). Google Scholar

[6]

O. DiekmannJ. 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

[7]

A. Fink, Almost Periodic Differential Equations, Springer, Berlin, 1974.  Google Scholar

[8]

A. B. GumelS. Ruan and T. Day, Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London Series B Biological Sciences, 271 (2020), 2223-2232.  doi: 10.1098/rspb.2004.2800.  Google Scholar

[9]

Z. D. GuoZ. Y. Wang and S. F. Zhang, Aerosol and surface distribution of severe acute respiratory syndrome coronavirus 2 in Hospital Wards, Wuhan, China, Emerg. Infect. Dis., 26 (2020), 1586-1591.  doi: 10.3201/eid2607.200885.  Google Scholar

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Providence, 1988.  Google Scholar

[11]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.  Google Scholar

[12]

X. HaoS. Cheng and D. Wu, Reconstruction of the full transmission dynamics of COVID-19 in Wuhan, Nature, 584 (2020), 420-424.  doi: 10.1038/s41586-020-2554-8.  Google Scholar

[13]

X. HeE. H. Y. Lau and P. Wu, Temporal dynamics in viral shedding and transmissibility of COVID-19, Nat Med, 26 (2020), 672-675.  doi: 10.1038/s41591-020-0869-5.  Google Scholar

[14]

J. HellewellS. Abbott and A. Gimma, Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, The Lancet Global Health, 8 (2020), 488-496.  doi: 10.1016/S2214-109X(20)30074-7.  Google Scholar

[15]

C. HuangY. Wang and X. Li, Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, The Lancet, 395 (2020), 497-506.  doi: 10.1016/S0140-6736(20)30183-5.  Google Scholar

[16]

G. M. HwangP. J. Mahoney and J. H. James, A model-based tool to predict the propagation of infectious disease via airports, Travel Medicine Infectious Disease, 10 (2020), 32-42.   Google Scholar

[17]

S. M. KisslerC. Tedijanto and E. Goldstein, Projecting the transmission dynamics of SARS-CoV-2 through the post-pandemic period, Science, 368 (2020), 860-868.  doi: 10.1126/science.abb5793.  Google Scholar

[18]

J. S. LavineM. Poss and B. T. Grenfell, Directly transmitted viral diseases: modeling the dynamics of transmission, Trends Microbiol, 16 (2020), 165-172.  doi: 10.1016/j.tim.2008.01.007.  Google Scholar

[19]

J. LeeG. Chowell and E. Jung, A dynamic compartmental model for the Middle East respiratory syndrome outbreak in the Republic of Korea: A retrospective analysis on control interventions and superspreading events, J. Theor. Biol., 408 (2016), 118-126.  doi: 10.1016/j.jtbi.2016.08.009.  Google Scholar

[20]

Y. LiuZ. Ning and Y. Chen, Aerodynamic analysis of SARS-CoV-2 in two Wuhan hospitals, Nature, 582 (2020), 557-560.  doi: 10.1038/s41586-020-2271-3.  Google Scholar

[21]

L. MarcC. Ted and C. Ben, Transmission dynamics and control of severe acute respiratory syndrome, Science, 300 (2003), 1966-1970.   Google Scholar

[22]

V. J. MunsterM. Koopmans and N. van Doremalen, A Novel coronavirus emerging in China-key questions for impact assessment, N. Engl. J. Med., 382 (2020), 692-694.   Google Scholar

[23]

R. A. Neher, R. Dyrdak and V. Druelle et al., Potential impact of seasonal forcing on a SARS-CoV-2 pandemic, Schwzerische medizinische Wochenschrift, 150 (2020). Google Scholar

[24]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Differ. Equ., 25 (2013), 1201-1231.   Google Scholar

[25]

A. Pan, L. Liu and C. Wang et al., Association of Public Health Interventions With the Epidemiology of the COVID-19 Outbreak in Wuhan, China, JAMA, (2020), 9 pp. Google Scholar

[26]

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.   Google Scholar

[27]

L. QiangB. G. Wang and X. Q. Zhao, Basic reproduction ratios for almost periodic compartmental models with time delay, J. Differ. Equ., 269 (2020), 4440-4476.   Google Scholar

[28]

J. Read, J. Bridgen and D. Cummings et al., Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions, Phil. Trans. R. Soc. B, 376 (2021), 9 pp. doi: 10.1098/rstb.2020.0265.  Google Scholar

[29]

H. L. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical society, Providence, 1995. Google Scholar

[30]

B. Tang, X. Wang and Q. Li et al., Estimation of the Transmission Risk of 2019-nCov and Its Implication for Public Health Interventions. Social ence Electronic Publishing, 2020. Google Scholar

[31]

Z. TongA. Tang and K. Li, Potential presymptomatic transmission of SARS-CoV-2, Zhejiang Province, China, 2020, Emerg. Infect. Dis., 26 (2020), 1052-1054.  doi: 10.3201/eid2605.200198.  Google Scholar

[32]

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

[33]

S. WangY. Liu and M. Liu, Research progress of basic regeneration number in COVID-19, Chinese Science Bulletin, 65 (2020), 2334-2341.  doi: 10.1360/TB-2020-0413.  Google Scholar

[34]

WHO, Middle East respiratory syndrome coronavirus (MERS-CoV). Google Scholar

[35]

WHO, Statement on the second meeting of the International Health Regulations (2005) Emergency Committee regarding the outbreak of novel coronavirus (2019-nCoV), 30 January 2020 Statement Geneva, Switzerland. Google Scholar

[36]

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

[37]

J. T. WuK. Leung and G. M. Leung, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, The Lancet, 395 (2020), 689-697.  doi: 10.1016/S0140-6736(20)30260-9.  Google Scholar

[38]

Z. YangZ. Zeng and K. Wang, Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions, J. Thorac. Dis., 12 (2020), 165-174.  doi: 10.21037/jtd.2020.02.64.  Google Scholar

[39]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2017. Google Scholar

[40]

N. ZhuD. Zhang and W. Wang, For the China Novel Coronavairus Investigating and Reaseach Team. A Novel coronavirus from patients with pneumonia in China, 2019., N. Engl. J. Med., 382 (2020), 727-733.   Google Scholar

show all references

References:
[1]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.  Google Scholar

[2]

Y. BaiL. Yao and T. Wei, Presumed asymptomatic carrier transmission of COVID-19, JAMA, 323 (2020), 790-808.  doi: 10.1001/jama.2020.2565.  Google Scholar

[3]

J. F. ChanS. Yuan and K. H. Kok, A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-person transmission: a study of a family cluster, The Lancet, 395 (2020), 514-523.  doi: 10.1016/S0140-6736(20)30154-9.  Google Scholar

[4]

C. Corduneanu, Almost Periodic Functions, New York, 1994. Google Scholar

[5]

COVID-19 prevention and control expert group of Chinese preventive medicine association. The latest understanding of epidemiological characteristics in COVID-19, Chinese J. Viral Diseases, 10 (2003), 86–92(Chinese). Google Scholar

[6]

O. DiekmannJ. 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

[7]

A. Fink, Almost Periodic Differential Equations, Springer, Berlin, 1974.  Google Scholar

[8]

A. B. GumelS. Ruan and T. Day, Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London Series B Biological Sciences, 271 (2020), 2223-2232.  doi: 10.1098/rspb.2004.2800.  Google Scholar

[9]

Z. D. GuoZ. Y. Wang and S. F. Zhang, Aerosol and surface distribution of severe acute respiratory syndrome coronavirus 2 in Hospital Wards, Wuhan, China, Emerg. Infect. Dis., 26 (2020), 1586-1591.  doi: 10.3201/eid2607.200885.  Google Scholar

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Providence, 1988.  Google Scholar

[11]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.  Google Scholar

[12]

X. HaoS. Cheng and D. Wu, Reconstruction of the full transmission dynamics of COVID-19 in Wuhan, Nature, 584 (2020), 420-424.  doi: 10.1038/s41586-020-2554-8.  Google Scholar

[13]

X. HeE. H. Y. Lau and P. Wu, Temporal dynamics in viral shedding and transmissibility of COVID-19, Nat Med, 26 (2020), 672-675.  doi: 10.1038/s41591-020-0869-5.  Google Scholar

[14]

J. HellewellS. Abbott and A. Gimma, Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, The Lancet Global Health, 8 (2020), 488-496.  doi: 10.1016/S2214-109X(20)30074-7.  Google Scholar

[15]

C. HuangY. Wang and X. Li, Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, The Lancet, 395 (2020), 497-506.  doi: 10.1016/S0140-6736(20)30183-5.  Google Scholar

[16]

G. M. HwangP. J. Mahoney and J. H. James, A model-based tool to predict the propagation of infectious disease via airports, Travel Medicine Infectious Disease, 10 (2020), 32-42.   Google Scholar

[17]

S. M. KisslerC. Tedijanto and E. Goldstein, Projecting the transmission dynamics of SARS-CoV-2 through the post-pandemic period, Science, 368 (2020), 860-868.  doi: 10.1126/science.abb5793.  Google Scholar

[18]

J. S. LavineM. Poss and B. T. Grenfell, Directly transmitted viral diseases: modeling the dynamics of transmission, Trends Microbiol, 16 (2020), 165-172.  doi: 10.1016/j.tim.2008.01.007.  Google Scholar

[19]

J. LeeG. Chowell and E. Jung, A dynamic compartmental model for the Middle East respiratory syndrome outbreak in the Republic of Korea: A retrospective analysis on control interventions and superspreading events, J. Theor. Biol., 408 (2016), 118-126.  doi: 10.1016/j.jtbi.2016.08.009.  Google Scholar

[20]

Y. LiuZ. Ning and Y. Chen, Aerodynamic analysis of SARS-CoV-2 in two Wuhan hospitals, Nature, 582 (2020), 557-560.  doi: 10.1038/s41586-020-2271-3.  Google Scholar

[21]

L. MarcC. Ted and C. Ben, Transmission dynamics and control of severe acute respiratory syndrome, Science, 300 (2003), 1966-1970.   Google Scholar

[22]

V. J. MunsterM. Koopmans and N. van Doremalen, A Novel coronavirus emerging in China-key questions for impact assessment, N. Engl. J. Med., 382 (2020), 692-694.   Google Scholar

[23]

R. A. Neher, R. Dyrdak and V. Druelle et al., Potential impact of seasonal forcing on a SARS-CoV-2 pandemic, Schwzerische medizinische Wochenschrift, 150 (2020). Google Scholar

[24]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Differ. Equ., 25 (2013), 1201-1231.   Google Scholar

[25]

A. Pan, L. Liu and C. Wang et al., Association of Public Health Interventions With the Epidemiology of the COVID-19 Outbreak in Wuhan, China, JAMA, (2020), 9 pp. Google Scholar

[26]

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.   Google Scholar

[27]

L. QiangB. G. Wang and X. Q. Zhao, Basic reproduction ratios for almost periodic compartmental models with time delay, J. Differ. Equ., 269 (2020), 4440-4476.   Google Scholar

[28]

J. Read, J. Bridgen and D. Cummings et al., Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions, Phil. Trans. R. Soc. B, 376 (2021), 9 pp. doi: 10.1098/rstb.2020.0265.  Google Scholar

[29]

H. L. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical society, Providence, 1995. Google Scholar

[30]

B. Tang, X. Wang and Q. Li et al., Estimation of the Transmission Risk of 2019-nCov and Its Implication for Public Health Interventions. Social ence Electronic Publishing, 2020. Google Scholar

[31]

Z. TongA. Tang and K. Li, Potential presymptomatic transmission of SARS-CoV-2, Zhejiang Province, China, 2020, Emerg. Infect. Dis., 26 (2020), 1052-1054.  doi: 10.3201/eid2605.200198.  Google Scholar

[32]

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

[33]

S. WangY. Liu and M. Liu, Research progress of basic regeneration number in COVID-19, Chinese Science Bulletin, 65 (2020), 2334-2341.  doi: 10.1360/TB-2020-0413.  Google Scholar

[34]

WHO, Middle East respiratory syndrome coronavirus (MERS-CoV). Google Scholar

[35]

WHO, Statement on the second meeting of the International Health Regulations (2005) Emergency Committee regarding the outbreak of novel coronavirus (2019-nCoV), 30 January 2020 Statement Geneva, Switzerland. Google Scholar

[36]

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

[37]

J. T. WuK. Leung and G. M. Leung, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, The Lancet, 395 (2020), 689-697.  doi: 10.1016/S0140-6736(20)30260-9.  Google Scholar

[38]

Z. YangZ. Zeng and K. Wang, Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions, J. Thorac. Dis., 12 (2020), 165-174.  doi: 10.21037/jtd.2020.02.64.  Google Scholar

[39]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2017. Google Scholar

[40]

N. ZhuD. Zhang and W. Wang, For the China Novel Coronavairus Investigating and Reaseach Team. A Novel coronavirus from patients with pneumonia in China, 2019., N. Engl. J. Med., 382 (2020), 727-733.   Google Scholar

Figure 3.  The variation of basic reproduction number, effective reproduction number and natural reproduction number
Figure 1.  The variation of model parameters
Figure 2.  Prediction and surveillance of new infected cases and total infected cases
[1]

Jorge Rebaza. On a model of COVID-19 dynamics. Electronic Research Archive, 2021, 29 (2) : 2129-2140. doi: 10.3934/era.2020108

[2]

Hailiang Liu, Xuping Tian. Data-driven optimal control of a seir model for COVID-19. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021093

[3]

Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1407-1424. doi: 10.3934/mbe.2017073

[4]

Tailei Zhang, Zhimin Li. Analysis of COVID-19 epidemic transmission trend based on a time-delayed dynamic model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021088

[5]

John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021, 8 (3) : 167-188. doi: 10.3934/jdg.2021004

[6]

Tao Zheng, Yantao Luo, Xinran Zhou, Long Zhang, Zhidong Teng. Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021154

[7]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, 2021, 20 (2) : 755-762. doi: 10.3934/cpaa.2020288

[8]

Yangjun Ma, Maoxing Liu, Qiang Hou, Jinqing Zhao. Modelling seasonal HFMD with the recessive infection in Shandong, China. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1159-1171. doi: 10.3934/mbe.2013.10.1159

[9]

Xuan Tian, Shangjiang Guo, Zhisu Liu. Qualitative analysis of a diffusive SEIR epidemic model with linear external source and asymptomatic infection in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021173

[10]

Mariantonia Cotronei, Tomas Sauer. Full rank filters and polynomial reproduction. Communications on Pure & Applied Analysis, 2007, 6 (3) : 667-687. doi: 10.3934/cpaa.2007.6.667

[11]

Oren Barnea, Rami Yaari, Guy Katriel, Lewi Stone. Modelling seasonal influenza in Israel. Mathematical Biosciences & Engineering, 2011, 8 (2) : 561-573. doi: 10.3934/mbe.2011.8.561

[12]

M. Predescu, R. Levins, T. Awerbuch-Friedlander. Analysis of a nonlinear system for community intervention in mosquito control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 605-622. doi: 10.3934/dcdsb.2006.6.605

[13]

Gechun Liang, Wei Wei. Optimal switching at Poisson random intervention times. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1483-1505. doi: 10.3934/dcdsb.2016008

[14]

Martin Luther Mann Manyombe, Joseph Mbang, Jean Lubuma, Berge Tsanou. Global dynamics of a vaccination model for infectious diseases with asymptomatic carriers. Mathematical Biosciences & Engineering, 2016, 13 (4) : 813-840. doi: 10.3934/mbe.2016019

[15]

Eunha Shim, Zhilan Feng, Carlos Castillo-Chavez. Differential impact of sickle cell trait on symptomatic and asymptomatic malaria. Mathematical Biosciences & Engineering, 2012, 9 (4) : 877-898. doi: 10.3934/mbe.2012.9.877

[16]

Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773

[17]

Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711

[18]

Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169

[19]

Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021006

[20]

Dmitry Krachun, Zhi-Wei Sun. On sums of four pentagonal numbers with coefficients. Electronic Research Archive, 2020, 28 (1) : 559-566. doi: 10.3934/era.2020029

[Back to Top]