American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Uniqueness and sign properties of minimizers in a quasilinear indefinite problem
  • CPAA Home
  • This Issue
  • Next Article
    Asymptotic behavior of eigenvalues of the Maxwell system in the presence of small changes in the interface of an inclusion
doi: 10.3934/cpaa.2021154
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.

Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence

Tao Zheng , Yantao Luo , Xinran Zhou , Long Zhang , and  Zhidong Teng

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

* Corresponding author

Received  February 2021 Revised  May 2021 Early access September 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11861065, 11771373, 11702237), the Open Project of Key Laboratory of Applied Mathematics of Xinjiang Province (Grant No. 2021D04014), the Natural Science Foundation of Xinjiang Province of China (Grant Nos. 2019D01C076, 2017D01C082), the Scientific Research Programmes of Colleges in Xinjiang (Grant No. XJEDU2021I002, XJEDU2021Y001), The Tianshan Youth Program-Training Program for Excellent Young Scientific and Technological Talents of Xinjiang (Grant No. 2019Q017)

A diffusion SEIAR model with Beddington-DeAngelis type incidence is proposed to characterize the spread of COVID-19 with spatial transmission. First, the well-posedness of solution is studied. Second, the basic reproduction number $ \mathcal R_{0} $ is derived and served as a threshold value to determine whether COVID-19 will spread. Meanwhile, we consider the effect of diffusion on the spread of COVID-19 in spatial homogenous environment, by which we can obtain that if $ \mathcal R_{0}<1 $, then the infection-free steady state is globally asymptotically stable, while if $ \mathcal R_{0}>1 $, then the endemic steady state is globally asymptotically stable. Furthermore, according to the official reporting data about COVID-19 in Wuhan, China, the actual value of $ \mathcal R_{0} $ is estimated, and comparing with other types of incidence, we find that the estimated peak with Beddington-DeAngelis type incidence is more close to the cases in reality. Finally, by numerical simulations, we can see that the diffusion behavior has evident impact on the spread of COVID-19 in spatial heterogeneity than homogeneity of environment.

Keywords: COVID-19, reaction-diffusion, basic reproduction number, Beddington-DeAngelis type incidence.
Mathematics Subject Classification: Primary: 34D23; Secondary: 34D05.
Citation: 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 and Applied Analysis, doi: 10.3934/cpaa.2021154
References:
[1]

T. Alberti and D. Faranda, On the uncertainty of real-time predictions of epidemic growths: A COVID-19 case study for China and Italy, Commun. Nonlinear Sci. Numer. Simul., 90 (2020), 105372.  doi: 10.1016/j.cnsns.2020.105372.

[2]

J. R. Beddington, Mutual interference between parasites or predators and its efect on searching efciency, J. Anim. Ecol., 44 (1975), 331-340. 

[3]

H. Berestycki, J. M. Roquejoffre and L. Rossi, Propagation of epidemics along lines with fast diffusion, Bull. Math. Biol., 81 (2021), 1-34.  doi: 10.1007/s11538-020-00826-8.

[4]

V. Capasso and G. Serio, A generalization of Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 41-61. 

[5]

M. Chinazzi, J. T. Davis and M. Ajelli, The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak, Science, 368 (2020), 395-400.  doi: 10.1126/science.aba9757.

[6]

J. F. W. Chan, S. 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, Lancet, 395 (2020), 514-523.  doi: 10.1016/S0140-6736(20)30154-9.

[7]

T. Chen, J. Rui and Q. Wang, et al., A mathematical model for simulating the phase-based transmissibility of a novel coronavirus, Infect. Dis. Poverty, 9 (2020), 8 pp. doi: 10.1186/s40249-020-00640-3.

[8]

V. Capasso, Mathematical Structures of Epidemic Systems, Springer-Verlag, 1993.

[9]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. 

[10] L. S. Chen and J. Chen, Nonlinear Biological Dynamics System, Scientific Press, China, 1993. 
[11]

Y. Cai, Z. Ding and B. Yang, Transmission dynamics of Zika virus with spatial structure-A case study in Rio de Janeiro, Brazil, Phys. A, 514 (2019), 729-740.  doi: 10.1016/j.physa.2018.09.100.

[12]

Y. Cai, X. Lian, Z. Peng and W. Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Anal. Real World Appl., 46 (2019), 178-194.  doi: 10.1016/j.nonrwa.2018.09.006.

[13]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892. 

[14]

N. T. Dieu, Asymptotic properties of a stochastic SIR epidemic model with Beddington-DeAngelis incidence rate, J. Dyn. Differ. Equ., 30 (2018), 93-106.  doi: 10.1007/s10884-016-9532-8.

[15]

W. Guan, Z. Ni and Y. Hu, Clinical characteristics of coronavirus disease 2019 in China, New Engl. J. Med., 382 (2020), 1708-1720.  doi: 10.1056/NEJMoa2002032.

[16]

G. Giordano, F. Blanchini and R. Bruno, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nat. Med., 26 (2020), 855-860.  doi: 10.1038/s41591-020-0883-7.

[17]

R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover. Public. Inc, Mineola, 1996.

[18]

J. Groeger, Divergence theorems and the supersphere, J. Geom. Phys., 77 (2014), 13-29. 

[19]

E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit, Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75 (1994), 17-29. 

[20]

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

[21]

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

[22]

L. Hu and L. Nie, Dynamic modeling and analysis of COVID-19 in different transmission process and control strategies, Math. Method. Appl. Sci., 44 (2021), 1409-1422.  doi: 10.1002/mma.6839.

[23]

Z. Jiang and J. Wei, Stability and bifurcation analysis in a delayed SIR model, Chaos Soliton. Fract., 35 (2008), 609-619.  doi: 10.1016/j.chaos.2006.05.045.

[24]

T. W. Joseph, L. Kathy and M. L. Gabriel, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, Lancet, 395 (2020), 689-697.  doi: 10.1016/S0140-6736(20)30260-9.

[25]

M. A. Khan and A. Atangana, Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alexandria Eng. J., 17 (2020), 2708-2724.  doi: 10.1016/j.aej.2020.02.033.

[26]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.

[27]

A. Kaddar, Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Anal. Model. Control., 15 (2010), 299-306.  doi: 10.15388/NA.15.3.14325.

[28]

S. Lai, N. W. Ruktanonchai and L. Zhou, Effect of nonpharmaceutical interventions to contain COVID-19 in China, Nature, 585 (2020), 410-413.  doi: 10.1038/s41586-020-2293-x.

[29]

X. Luo, S. Feng and J. Yang, et al., Analysis of potential risk of COVID-19 infections in China based on a pairwise epidemic model, preprint.

[30]

Y. Luo, L. Zhang, T. Zheng and Z. D. Teng, Analysis of a diffusive virus infection model with humoral immunity, cell-to-cell transmission and nonlinear incidence, Phys. A, 535 (2019), 122415.  doi: 10.1016/j.physa.2019.122415.

[31]

Y. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.

[32]

Y. Mammeri, A reaction-diffusion system to better comprehend the unlockdown: Application of SEIR-type model with diffusion to the spatial spread of COVID-19 in France, Comput. Math. Biophys., 8 2020), 102–113. doi: 10.1515/cmb-2020-0104.

[33]

H. Nishiura, N. M. Linton and A. R. Akhmetzhanov, Serial interval of novel coronavirus (COVID-19) infections, Int. J. Infect. Dis., 93 (2020), 284-286.  doi: 10.1016/j.ijid.2020.02.060.

[34]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equ., 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.

[35]

X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, J. Math. Biol., 76 (2018), 1831-1872.  doi: 10.1007/s00285-017-1202-x.

[36]

X. Sun, X. Huo and J. Wu, Simulation study about large-scale use of convalescent plasma therapy for the treatment of COVID-19 Patients with Critical symptoms, Acta Math. Appl. Sin. (Chin. Ser.), 43 (2020), 211-226. 

[37]

P. Song, Y. Lou and L. Zhu, Multi-stage and multi-scale patch model and the case study of novel coronavirus, Acta Math. Appl. Sin. (Chin. Ser.), 43 (2020), 174-199. 

[38]

P. Song, Y. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differ. Equ., 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.

[39]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soci., 1995.

[40]

H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. Theory Methods Appl., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[41]

B. Tang, X. Wang and Q. Li, et al., Estimation of the transmission risk of 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 13 pp. doi: 10.3390/jcm9020462.

[42]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichoyomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.

[43]

A. Viguerie, G. Lorenzo and F. Auricchio, Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion, Appl. Math. Lett., 111 (2021), 106617.  doi: 10.1016/j.aml.2020.106617.

[44]

X. Wang, Y. Tao and X. Song, Delayed HIV-1 infection model Beddington-DeAngelis functional response, Nonlinear Dyn., 62 (2010), 67-72.  doi: 10.1007/s11071-010-9699-1.

[45]

J. Wang, F. Xie and T. Kuniya, Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104951.  doi: 10.1016/j.cnsns.2019.104951.

[46]

H. Wang and N. Yamamoto, Using a partial differential equation with Google Mobility data to predict COVID-19 in Arizona, Math. Biosci. Eng., 17 (2020), 4891-4904.  doi: 10.3934/mbe.2020266.

[47]

M. Wang, Nonlinear Elliptic Equations, Science Public, Beijing, 2010.

[48]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.

[49]

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.

[50]

L. Xue, S. Jing and J. C. Miller, A data-driven network model for the emerging COVID-19 epidemics in Wuhan, Toronto and Italy, Math. Biosci., 326 (2020), 108391.  doi: 10.1016/j.mbs.2020.108391.

[51]

R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Soliton Fract., 41 (2009), 2319-2325.  doi: 10.1016/j.chaos.2008.09.007.

[52]

C. Yang and J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708-2724.  doi: 10.3934/mbe.2020148.

[53]

P. Zhou, X. Yang and X. Wang, et al., Discovery of a novel coronavirs associated with the recent pneumonia outbreak in humans and its potential bat origin, BioRxiv, (2020). Available from: https://doi.org/10.1101/2020.01.22.914952.

[54]

W. Zhou, A. Wang and F. Xia, Effects of media reporting on mitigating spread of COVID-19 in the early phase of the outbreak, Math. Biosci. Eng., 17 (2020), 2693-2707.  doi: 10.3934/mbe.2020147.

[55]

L. Zhu, H. Zhao and X. Wang, Stability and bifurcation analysis in a delayed reaction-diffusion malware propagation model, Comput. Math. Appl., 69 (2015), 852-875.  doi: 10.1016/j.camwa.2015.02.004.

[56]

WHO COVID-19 Dashboard. Geneva: World Health Organization, 2020.

show all references

References:
[1]

T. Alberti and D. Faranda, On the uncertainty of real-time predictions of epidemic growths: A COVID-19 case study for China and Italy, Commun. Nonlinear Sci. Numer. Simul., 90 (2020), 105372.  doi: 10.1016/j.cnsns.2020.105372.

[2]

J. R. Beddington, Mutual interference between parasites or predators and its efect on searching efciency, J. Anim. Ecol., 44 (1975), 331-340. 

[3]

H. Berestycki, J. M. Roquejoffre and L. Rossi, Propagation of epidemics along lines with fast diffusion, Bull. Math. Biol., 81 (2021), 1-34.  doi: 10.1007/s11538-020-00826-8.

[4]

V. Capasso and G. Serio, A generalization of Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 41-61. 

[5]

M. Chinazzi, J. T. Davis and M. Ajelli, The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak, Science, 368 (2020), 395-400.  doi: 10.1126/science.aba9757.

[6]

J. F. W. Chan, S. 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, Lancet, 395 (2020), 514-523.  doi: 10.1016/S0140-6736(20)30154-9.

[7]

T. Chen, J. Rui and Q. Wang, et al., A mathematical model for simulating the phase-based transmissibility of a novel coronavirus, Infect. Dis. Poverty, 9 (2020), 8 pp. doi: 10.1186/s40249-020-00640-3.

[8]

V. Capasso, Mathematical Structures of Epidemic Systems, Springer-Verlag, 1993.

[9]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. 

[10] L. S. Chen and J. Chen, Nonlinear Biological Dynamics System, Scientific Press, China, 1993. 
[11]

Y. Cai, Z. Ding and B. Yang, Transmission dynamics of Zika virus with spatial structure-A case study in Rio de Janeiro, Brazil, Phys. A, 514 (2019), 729-740.  doi: 10.1016/j.physa.2018.09.100.

[12]

Y. Cai, X. Lian, Z. Peng and W. Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Anal. Real World Appl., 46 (2019), 178-194.  doi: 10.1016/j.nonrwa.2018.09.006.

[13]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892. 

[14]

N. T. Dieu, Asymptotic properties of a stochastic SIR epidemic model with Beddington-DeAngelis incidence rate, J. Dyn. Differ. Equ., 30 (2018), 93-106.  doi: 10.1007/s10884-016-9532-8.

[15]

W. Guan, Z. Ni and Y. Hu, Clinical characteristics of coronavirus disease 2019 in China, New Engl. J. Med., 382 (2020), 1708-1720.  doi: 10.1056/NEJMoa2002032.

[16]

G. Giordano, F. Blanchini and R. Bruno, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nat. Med., 26 (2020), 855-860.  doi: 10.1038/s41591-020-0883-7.

[17]

R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover. Public. Inc, Mineola, 1996.

[18]

J. Groeger, Divergence theorems and the supersphere, J. Geom. Phys., 77 (2014), 13-29. 

[19]

E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit, Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75 (1994), 17-29. 

[20]

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

[21]

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

[22]

L. Hu and L. Nie, Dynamic modeling and analysis of COVID-19 in different transmission process and control strategies, Math. Method. Appl. Sci., 44 (2021), 1409-1422.  doi: 10.1002/mma.6839.

[23]

Z. Jiang and J. Wei, Stability and bifurcation analysis in a delayed SIR model, Chaos Soliton. Fract., 35 (2008), 609-619.  doi: 10.1016/j.chaos.2006.05.045.

[24]

T. W. Joseph, L. Kathy and M. L. Gabriel, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, Lancet, 395 (2020), 689-697.  doi: 10.1016/S0140-6736(20)30260-9.

[25]

M. A. Khan and A. Atangana, Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alexandria Eng. J., 17 (2020), 2708-2724.  doi: 10.1016/j.aej.2020.02.033.

[26]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.

[27]

A. Kaddar, Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Anal. Model. Control., 15 (2010), 299-306.  doi: 10.15388/NA.15.3.14325.

[28]

S. Lai, N. W. Ruktanonchai and L. Zhou, Effect of nonpharmaceutical interventions to contain COVID-19 in China, Nature, 585 (2020), 410-413.  doi: 10.1038/s41586-020-2293-x.

[29]

X. Luo, S. Feng and J. Yang, et al., Analysis of potential risk of COVID-19 infections in China based on a pairwise epidemic model, preprint.

[30]

Y. Luo, L. Zhang, T. Zheng and Z. D. Teng, Analysis of a diffusive virus infection model with humoral immunity, cell-to-cell transmission and nonlinear incidence, Phys. A, 535 (2019), 122415.  doi: 10.1016/j.physa.2019.122415.

[31]

Y. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.

[32]

Y. Mammeri, A reaction-diffusion system to better comprehend the unlockdown: Application of SEIR-type model with diffusion to the spatial spread of COVID-19 in France, Comput. Math. Biophys., 8 2020), 102–113. doi: 10.1515/cmb-2020-0104.

[33]

H. Nishiura, N. M. Linton and A. R. Akhmetzhanov, Serial interval of novel coronavirus (COVID-19) infections, Int. J. Infect. Dis., 93 (2020), 284-286.  doi: 10.1016/j.ijid.2020.02.060.

[34]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equ., 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.

[35]

X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, J. Math. Biol., 76 (2018), 1831-1872.  doi: 10.1007/s00285-017-1202-x.

[36]

X. Sun, X. Huo and J. Wu, Simulation study about large-scale use of convalescent plasma therapy for the treatment of COVID-19 Patients with Critical symptoms, Acta Math. Appl. Sin. (Chin. Ser.), 43 (2020), 211-226. 

[37]

P. Song, Y. Lou and L. Zhu, Multi-stage and multi-scale patch model and the case study of novel coronavirus, Acta Math. Appl. Sin. (Chin. Ser.), 43 (2020), 174-199. 

[38]

P. Song, Y. Lou and Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differ. Equ., 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.

[39]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soci., 1995.

[40]

H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. Theory Methods Appl., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[41]

B. Tang, X. Wang and Q. Li, et al., Estimation of the transmission risk of 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 13 pp. doi: 10.3390/jcm9020462.

[42]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichoyomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.

[43]

A. Viguerie, G. Lorenzo and F. Auricchio, Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion, Appl. Math. Lett., 111 (2021), 106617.  doi: 10.1016/j.aml.2020.106617.

[44]

X. Wang, Y. Tao and X. Song, Delayed HIV-1 infection model Beddington-DeAngelis functional response, Nonlinear Dyn., 62 (2010), 67-72.  doi: 10.1007/s11071-010-9699-1.

[45]

J. Wang, F. Xie and T. Kuniya, Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104951.  doi: 10.1016/j.cnsns.2019.104951.

[46]

H. Wang and N. Yamamoto, Using a partial differential equation with Google Mobility data to predict COVID-19 in Arizona, Math. Biosci. Eng., 17 (2020), 4891-4904.  doi: 10.3934/mbe.2020266.

[47]

M. Wang, Nonlinear Elliptic Equations, Science Public, Beijing, 2010.

[48]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.

[49]

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.

[50]

L. Xue, S. Jing and J. C. Miller, A data-driven network model for the emerging COVID-19 epidemics in Wuhan, Toronto and Italy, Math. Biosci., 326 (2020), 108391.  doi: 10.1016/j.mbs.2020.108391.

[51]

R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Soliton Fract., 41 (2009), 2319-2325.  doi: 10.1016/j.chaos.2008.09.007.

[52]

C. Yang and J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708-2724.  doi: 10.3934/mbe.2020148.

[53]

P. Zhou, X. Yang and X. Wang, et al., Discovery of a novel coronavirs associated with the recent pneumonia outbreak in humans and its potential bat origin, BioRxiv, (2020). Available from: https://doi.org/10.1101/2020.01.22.914952.

[54]

W. Zhou, A. Wang and F. Xia, Effects of media reporting on mitigating spread of COVID-19 in the early phase of the outbreak, Math. Biosci. Eng., 17 (2020), 2693-2707.  doi: 10.3934/mbe.2020147.

[55]

L. Zhu, H. Zhao and X. Wang, Stability and bifurcation analysis in a delayed reaction-diffusion malware propagation model, Comput. Math. Appl., 69 (2015), 852-875.  doi: 10.1016/j.camwa.2015.02.004.

[56]

WHO COVID-19 Dashboard. Geneva: World Health Organization, 2020.

Figure 1.  An illustration of the model (1.2) describing the transmission of COVID-19 infection
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The fitting results of cumulative confirmed cases from Jan 23rd to Feb 11rd, 2020. The blue curve denotes fitting curve of model (8.1). The star denotes the real data of cumulative confirmed cases
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Fig. (a) is a simulation result for the outbreak size in Wuhan using model (8.1), the parameters from Table 1. Fig. (b) and (c) is the effects of different recovery rate and transmission rate on disease dynamics in symptomatic infected $ I(t) $, respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The effect of different incidence rates on the peak number of asymptomatic infected persons $ I(t) $ in the short term
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Spatial distribution of the number of symptomatic infections $ I(x, t) $ in the short term for $ d = 1.25\times 10^{-2} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The short time behaviour of the solution $ I(x, t) $ of model (7.1) with $ d = 0, 1.25\times 10^{-3} $, all other parameters as Table 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The long time behaviour of the solution $ I(x, t) $ of model (7.1) with $ (\beta, \alpha, \eta) = (6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $ and $ d = 1.25\times 10^{-2} $, all other parameters are shown in Table 1 $ (\mathcal{R}_{0} = 4.1062>1) $. COVID-19 disease is finally reaching persistence
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  The long time behaviour of the solution $ I(x, t) $ of model (7.1) with $ (\beta, \alpha, \eta) = (1.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $ and $ d = 1.25\times 10^{-2} $, all other parameters are shown in Table 1 $ (\mathcal{R}_{0} = 0.9551<1) $. COVID-19 disease is finally heading towards extinction
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Spatial distribution of the number of symptomatic infections $ I(x, t) $ in the short term for model (1.2)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  The short time behaviour of the solution $ I(x, t) $ of model (1.2) with $ (\beta, \alpha, \eta) = (1+0.5\cos2\pi x)(6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $ and $ d = 0, 1.25\times 10^{-3}, 1.25\times 10^{-2} $, all other parameters are shown in Table 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11.  The long time behaviour of the solution $ I(x, t) $ of model (1.2) with $ (\beta(x), \alpha(x), \eta(x)) = (1+0.5\cos2\pi x)(6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $ and $ d = 1.25\times 10^{-2} $, all other parameters are shown in Table 1 $ (\mathcal{R}_{0} = 4.2476>1) $. COVID-19 disease is finally reaching persistence. Fig (b) is the relation between $ \mathcal{R}_{0} $ and $ c $ in $ (\beta(x), \alpha(x), \eta(x)) = (1+c\cos 2\pi x)(6.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12.  The long time behaviour of the solution $ I(x, t) $ of model (1.2) with $ (\beta(x), \alpha(x), \eta(x)) = (1+0.5\cos2\pi x)(1.10\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $ and $ d = 1.25\times 10^{-2} $, all other parameters are shown in Table 1 $ (\mathcal{R}_{0} = 0.7231<1) $. COVID-19 disease is finally heading towards extinction. Fig (b) is the relation between $ \mathcal{R}_{0} $ and $ c $ in $ (\beta(x), \alpha(x), \eta(x)) = (1+c\cos 2\pi x)(1.10\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 13.  The basic reproduction number $ \mathcal{R}_{0} $ of model (1.2) for $ 0\leq c\leq1 $ and $ k = 2, 7, 13 $, where $ (\beta(x), \alpha(x), \eta(x)) = (1+c\cos k\pi x)(1.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $ and $ d = 1.25\times 10^{-2} $, all other parameters are shown in Table 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 14.  The solution of $ I(x, t) $ of model (7.1) with initial condition (8.2) and $ d = 0, \ 1.25\times 10^{-3}, \ 1.25\times 10^{-2} $, where $ (\beta, \alpha, \eta) = (1.51\times 10^{-7}, 3.11\times 10^{-7}, 1.56\times 10^{-7}) $, all other parameters are shown in Table 1
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  The values of parameters for model (8.1)
Parameter Description Value Reference
$ \lambda $ Influx rate $ \frac{8, 999, 990}{76.79\times365} $ [52]
$ \mu $ Natural mortality $ \frac{1}{76.79\times365} $ [25]
$ \beta $ Infection rate between $ S $ and $ E $ $ 6.51\times10^{-7} $ Fitted
$ \alpha $ Infection rate between $ S $ and $ I $ $ 3.11\times10^{-7} $ [41]
$ \eta $ Infection rate between $ S $ and $ A $ $ 1.56\times10^{-7} $ [41]
$ \omega $ Incubation period 1/7 [41]
$ \theta $ Incubation period 1/7 [41]
$ \delta $ Asymptomatic infection rate 0.2412 Fitted
$ \gamma $ Recovery rate of $ I $ 1/15 [41]
$ \varpi $ Recovery rate of $ A $ 0.8613 Fitted
$ m_{1} $ Inhibition rate $ 1.00\times10^{-6} $ Fitted
$ m_{2} $ Inhibition rate $ 7.11\times10^{-4} $ Fitted
$ n_{1} $ Inhibition rate $ 8.92\times10^{-4} $ Fitted
$ n_{2} $ Inhibition rate $ 1.00\times10^{-3} $ Fitted
$ e_{1} $ Inhibition rate $ 1.00\times10^{-3} $ Fitted
$ e_{2} $ Inhibition rate $ 7.83\times10^{-4} $ Fitted
$ S(0) $ Initial value 8998187 [52]
$ E(0) $ Initial value 845.6 Fitted
$ I(0) $ Initial value 475 [52]
$ A(0) $ Initial value 472.4 Fitted
$ R(0) $ Initial value 10 [52]
Table Options

Download as excel

Parameter Description Value Reference
$ \lambda $ Influx rate $ \frac{8, 999, 990}{76.79\times365} $ [52]
$ \mu $ Natural mortality $ \frac{1}{76.79\times365} $ [25]
$ \beta $ Infection rate between $ S $ and $ E $ $ 6.51\times10^{-7} $ Fitted
$ \alpha $ Infection rate between $ S $ and $ I $ $ 3.11\times10^{-7} $ [41]
$ \eta $ Infection rate between $ S $ and $ A $ $ 1.56\times10^{-7} $ [41]
$ \omega $ Incubation period 1/7 [41]
$ \theta $ Incubation period 1/7 [41]
$ \delta $ Asymptomatic infection rate 0.2412 Fitted
$ \gamma $ Recovery rate of $ I $ 1/15 [41]
$ \varpi $ Recovery rate of $ A $ 0.8613 Fitted
$ m_{1} $ Inhibition rate $ 1.00\times10^{-6} $ Fitted
$ m_{2} $ Inhibition rate $ 7.11\times10^{-4} $ Fitted
$ n_{1} $ Inhibition rate $ 8.92\times10^{-4} $ Fitted
$ n_{2} $ Inhibition rate $ 1.00\times10^{-3} $ Fitted
$ e_{1} $ Inhibition rate $ 1.00\times10^{-3} $ Fitted
$ e_{2} $ Inhibition rate $ 7.83\times10^{-4} $ Fitted
$ S(0) $ Initial value 8998187 [52]
$ E(0) $ Initial value 845.6 Fitted
$ I(0) $ Initial value 475 [52]
$ A(0) $ Initial value 472.4 Fitted
$ R(0) $ Initial value 10 [52]
Table 2.  Peak values of symptomatic infection $ I(t) $ with different incidence functions
Transmission rate Forms Fitted parameters Peak value
Bilinear incidence $ \beta SE $, $ \alpha SI $, $ \eta SA $ $ \beta $, $ \delta $, $ \varpi $ 674950
Saturated incidence for the susceptible $ \frac{\beta SE}{1+m_{1}S} $, $ \frac{\beta SI}{1+n_{1}S} $, $ \frac{\beta SA}{1+e_{1}S} $ $ \beta $, $ m_{1} $, $ n_{1} $, $ e_{1} $, $ \delta $, $ \varpi $ 3357600
Saturated incidence for the infected $ \frac{\beta SE}{1+m_{2}E} $, $ \frac{\beta SI}{1+n_{2}I} $, $ \frac{\beta SA}{1+e_{2}A} $ $ \beta $, $ m_{2} $, $ n_{2} $, $ e_{2} $, $ \delta $, $ \varpi $ 71357
Beddington-DeAngelis type incidence $ \frac{\beta SE}{1+m_{1}S+m_{2}E} $, $ \frac{\beta SI}{1+n_{1}S+n_{2}I} $, $ \frac{\beta SA}{1+e_{1}S+e_{2}A} $ $ \beta $, $ m_{1} $, $ m_{2} $, $ n_{1} $, $ n_{2} $, $ e_{1} $, $ e_{2} $, $ \delta $, $ \varpi $ 67599
Table Options

Download as excel

Transmission rate Forms Fitted parameters Peak value
Bilinear incidence $ \beta SE $, $ \alpha SI $, $ \eta SA $ $ \beta $, $ \delta $, $ \varpi $ 674950
Saturated incidence for the susceptible $ \frac{\beta SE}{1+m_{1}S} $, $ \frac{\beta SI}{1+n_{1}S} $, $ \frac{\beta SA}{1+e_{1}S} $ $ \beta $, $ m_{1} $, $ n_{1} $, $ e_{1} $, $ \delta $, $ \varpi $ 3357600
Saturated incidence for the infected $ \frac{\beta SE}{1+m_{2}E} $, $ \frac{\beta SI}{1+n_{2}I} $, $ \frac{\beta SA}{1+e_{2}A} $ $ \beta $, $ m_{2} $, $ n_{2} $, $ e_{2} $, $ \delta $, $ \varpi $ 71357
Beddington-DeAngelis type incidence $ \frac{\beta SE}{1+m_{1}S+m_{2}E} $, $ \frac{\beta SI}{1+n_{1}S+n_{2}I} $, $ \frac{\beta SA}{1+e_{1}S+e_{2}A} $ $ \beta $, $ m_{1} $, $ m_{2} $, $ n_{1} $, $ n_{2} $, $ e_{1} $, $ e_{2} $, $ \delta $, $ \varpi $ 67599
[1]

Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170
[2]

Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations and Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115
[3]

Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170
[4]

Jorge Rebaza. On a model of COVID-19 dynamics. Electronic Research Archive, 2021, 29 (2) : 2129-2140. doi: 10.3934/era.2020108
[5]

Peng Yang, Yuanshi Wang. On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3845-3895. doi: 10.3934/dcdsb.2021209
[6]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035
[7]

Qi Luo, Ryan Weightman, Sean T. McQuade, Mateo Díaz, Emmanuel Trélat, William Barbour, Dan Work, Samitha Samaranayake, Benedetto Piccoli. Optimization of vaccination for COVID-19 in the midst of a pandemic. Networks and Heterogeneous Media, 2022, 17 (3) : 443-466. doi: 10.3934/nhm.2022016
[8]

Nicola Bellomo, Diletta Burini, Nisrine Outada. Multiscale models of Covid-19 with mutations and variants. Networks and Heterogeneous Media, 2022, 17 (3) : 293-310. doi: 10.3934/nhm.2022008
[9]

Hailiang Liu, Xuping Tian. Data-driven optimal control of a seir model for COVID-19. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021093
[10]

Emiliano Alvarez, Juan Gabriel Brida, Lucía Rosich, Erick Limas. Analysis of communities of countries with similar dynamics of the COVID-19 pandemic evolution. Journal of Dynamics and Games, 2022, 9 (1) : 75-96. doi: 10.3934/jdg.2021026
[11]

Haitao Song, Fang Liu, Feng Li, Xiaochun Cao, Hao Wang, Zhongwei Jia, Huaiping Zhu, Michael Y. Li, Wei Lin, Hong Yang, Jianghong Hu, Zhen Jin. Modeling the second outbreak of COVID-19 with isolation and contact tracing. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021294
[12]

Gabriel Illanes, Ernesto Mordecki, Andrés Sosa. On the impact of the Covid-19 health crisis on GDP forecasting: An empirical approach. Journal of Dynamics and Games, 2022  doi: 10.3934/jdg.2022008
[13]

Monique Chyba, Rinaldo M. Colombo, Mauro Garavello, Benedetto Piccoli. Advanced mathematical methodologies to contrast COVID-19 pandemic. Networks and Heterogeneous Media, 2022, 17 (3) : i-ii. doi: 10.3934/nhm.2022020
[14]

Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117
[15]

Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215
[16]

Renji Han, Binxiang Dai, Lin Wang. Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response. Mathematical Biosciences & Engineering, 2018, 15 (3) : 595-627. doi: 10.3934/mbe.2018027
[17]

Sze-Bi Hsu, Shigui Ruan, Ting-Hui Yang. On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2331-2353. doi: 10.3934/dcdsb.2013.18.2331
[18]

Seong Lee, Inkyung Ahn. Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses. Communications on Pure and Applied Analysis, 2017, 16 (2) : 427-442. doi: 10.3934/cpaa.2017022
[19]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
[20]

Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117

2021 Impact Factor: 1.273

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy