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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 |
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.
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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.
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|
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L. S. Chen and J. Chen, Nonlinear Biological Dynamics System, Scientific Press, China, 1993.
![]() |
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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. |
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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. |
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D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill,
A model for tropic interaction, Ecology, 56 (1975), 881-892.
|
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N. T. Dieu,
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Stability and bifurcation analysis in a delayed SIR model, Chaos Soliton. Fract., 35 (2008), 609-619.
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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.
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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. |
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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. |
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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. |
[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. |












Parameter | Description | Value | Reference |
Influx rate | [52] | ||
Natural mortality | [25] | ||
Infection rate between |
Fitted | ||
Infection rate between |
[41] | ||
Infection rate between |
[41] | ||
Incubation period | 1/7 | [41] | |
Incubation period | 1/7 | [41] | |
Asymptomatic infection rate | 0.2412 | Fitted | |
Recovery rate of |
1/15 | [41] | |
Recovery rate of |
0.8613 | Fitted | |
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Initial value | 8998187 | [52] | |
Initial value | 845.6 | Fitted | |
Initial value | 475 | [52] | |
Initial value | 472.4 | Fitted | |
Initial value | 10 | [52] |
Parameter | Description | Value | Reference |
Influx rate | [52] | ||
Natural mortality | [25] | ||
Infection rate between |
Fitted | ||
Infection rate between |
[41] | ||
Infection rate between |
[41] | ||
Incubation period | 1/7 | [41] | |
Incubation period | 1/7 | [41] | |
Asymptomatic infection rate | 0.2412 | Fitted | |
Recovery rate of |
1/15 | [41] | |
Recovery rate of |
0.8613 | Fitted | |
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Inhibition rate | Fitted | ||
Initial value | 8998187 | [52] | |
Initial value | 845.6 | Fitted | |
Initial value | 475 | [52] | |
Initial value | 472.4 | Fitted | |
Initial value | 10 | [52] |
Transmission rate | Forms | Fitted parameters | Peak value |
Bilinear incidence | 674950 | ||
Saturated incidence for the susceptible | 3357600 | ||
Saturated incidence for the infected | 71357 | ||
Beddington-DeAngelis type incidence | 67599 |
Transmission rate | Forms | Fitted parameters | Peak value |
Bilinear incidence | 674950 | ||
Saturated incidence for the susceptible | 3357600 | ||
Saturated incidence for the infected | 71357 | ||
Beddington-DeAngelis type incidence | 67599 |
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