doi: 10.3934/cpaa.2021154
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Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence

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.

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 &amp; Applied Analysis, doi: 10.3934/cpaa.2021154
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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.  Google Scholar

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Y. LuoL. ZhangT. 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.  Google Scholar

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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.  Google Scholar

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X. RenY. TianL. 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.  Google Scholar

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X. SunX. 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.   Google Scholar

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P. SongY. 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.   Google Scholar

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P. SongY. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[43]

A. ViguerieG. 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.  Google Scholar

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X. WangY. 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.  Google Scholar

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J. WangF. 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.  Google Scholar

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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.  Google Scholar

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

[48]

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

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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.  Google Scholar

[50]

L. XueS. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

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W. ZhouA. 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.  Google Scholar

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L. ZhuH. 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.  Google Scholar

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

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.  Google Scholar

[2]

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

[3]

H. BerestyckiJ. 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.  Google Scholar

[4]

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

[5]

M. ChinazziJ. 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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

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

Y. CaiZ. 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.  Google Scholar

[12]

Y. CaiX. LianZ. 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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

G. GiordanoF. 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.  Google Scholar

[17]

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

[18]

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

[19]

E. E. HolmesM. A. LewisJ. E. Banks and R. R. Veit, Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75 (1994), 17-29.   Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

T. W. JosephL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. LaiN. 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.  Google Scholar

[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. Google Scholar

[30]

Y. LuoL. ZhangT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

H. NishiuraN. 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.  Google Scholar

[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.  Google Scholar

[35]

X. RenY. TianL. 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.  Google Scholar

[36]

X. SunX. 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.   Google Scholar

[37]

P. SongY. 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.   Google Scholar

[38]

P. SongY. 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.  Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

A. ViguerieG. 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.  Google Scholar

[44]

X. WangY. 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.  Google Scholar

[45]

J. WangF. 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.  Google Scholar

[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.  Google Scholar

[47]

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

[48]

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

[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.  Google Scholar

[50]

L. XueS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[54]

W. ZhouA. 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.  Google Scholar

[55]

L. ZhuH. 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.  Google Scholar

[56]

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

Figure 1.  An illustration of the model (1.2) describing the transmission of COVID-19 infection
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
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 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 4.  The effect of different incidence rates on the peak number of asymptomatic infected persons $ I(t) $ in the short term
Figure 5.  Spatial distribution of the number of symptomatic infections $ I(x, t) $ in the short term for $ d = 1.25\times 10^{-2} $
Table 1">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
Table 1 $ (\mathcal{R}_{0} = 4.1062>1) $. COVID-19 disease is finally reaching persistence">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
Table 1 $ (\mathcal{R}_{0} = 0.9551<1) $. COVID-19 disease is finally heading towards extinction">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 9.  Spatial distribution of the number of symptomatic infections $ I(x, t) $ in the short term for model (1.2)
Table 1">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
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 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}) $
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 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}) $
Table 1">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
Table 1">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
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]
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
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
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