# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021150
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## Dynamical analysis in disease transmission and final epidemic size

 1 Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai, 264209, China 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

* Corresponding author

Received  January 2021 Revised  July 2021 Early access September 2021

Fund Project: Research partially supported by the Natural Sciences and Engineering Research Council of Canada, China Scholarship Council

We propose two compartment models to study the disease transmission dynamics, then apply the models to the current COVID-19 pandemic and to explore the potential impact of the interventions, and try to provide insights into the future health care demand. Starting with an SEAIQR model by combining the effect from exposure, asymptomatic and quarantine, then extending the model to the one with ages below and above 65 years old, and classify the infectious individuals according to their severity. We focus our analysis on each model with and without vital dynamics. In the models with vital dynamics, we study the dynamical properties including the global stability of the disease free equilibrium and the existence of endemic equilibrium, with respect to the basic reproduction number. Whereas in the models without vital dynamics, we address the final epidemic size rigorously, which is one of the common but difficult questions regarding an epidemic. Finally, we apply our models to estimate the basic reproduction number and the final epidemic size of disease by using the data of COVID-19 confirmed cases in Canada and Newfoundland & Labrador province.

Citation: Daifeng Duan, Cuiping Wang, Yuan Yuan. Dynamical analysis in disease transmission and final epidemic size. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2021150
##### References:
 [1] V. Andreasen, The final size of an epidemic and its relation to the basic reproduction number, Bull. Math. Biol., 73 (2011), 2305-2321.  doi: 10.1007/s11538-010-9623-3. [2] J. A. Backer, D. Klinkenberg and J. Wallinga, Incubation period of 2019 novel coronavirus (2019-nCoV) infections among travellers from Wuhan, China, Eurosurveillance, 25 (2020), 2000062. [3] M. V. Barbarossa, A. D$\mathrm{\acute{e}}$nes and G. Kiss, et al., Transmission dynamics and final epidemic size of Ebola virus disease outbreaks with varying interventions, PloS. one, 10(7), (2015). [4] P. Brasil, Jr. J. P. Pereira and M. E. Moreira, Zika virus infection in pregnant women in Rio de Janeiro, N. Engl. J. Med., 375 (2016), 2321-2334.  doi: 10.1056/NEJMoa1602412. [5] F. Brauer, Early estimates of epidemic final sizes, J. Biol. Dynam., 13 (2019), 23-30. doi: 10.1080/17513758.2018.1469792. [6] F. Brauer, The final size of a serious epidemic, Bull. Math. Biol., 81 (2019), 869-877.  doi: 10.1007/s11538-018-00549-x. [7] T. Britton, F. Ball and P. Trapman, A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2, Science, 369 (2020), 846-849.  doi: 10.1126/science.abc6810. [8] S. Y. Del Valle, J. M. Hyman and N. Chitnis, Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases, Math. Bio. Eng., 10 (2013), 1475-1497.  doi: 10.3934/mbe.2013.10.1475. [9] A. D$\mathrm{\acute{e}}$nes and A. B. Gumel, Modeling the impact of quarantine during an outbreak of Ebola virus disease, Infectious Disease Modelling, 4 (2019), 12-27. [10] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324. [11] T. House, J. V. Ross and D. Sirl, How big is an outbreak likely to be? Methods for epidemic final-size calculation, Proc. R. Soc. A., 469 (2013), 20120436.  doi: 10.1098/rspa.2012.0436. [12] N. Imai, I. Dorigatti and A. Cori, et al., Report 2: Estimating the potential total number of novel Coronavirus cases in Wuhan City, China (2020)., [13] R. M. Jasmer, P. Nahid and P. C. Hopewell, Latent tuberculosis infection, N. Engl. J. Med., 347 (2002), 1860-1866. [14] D. G. Lalloo, D. Shingadia and G. Pasvol, UK malaria treatment guidelines, J. Infect., 54 (2007), 111-121. [15] E. M. Leroy, B. Kumulungui and X. Pourrut, Fruit bats as reservoirs of Ebola virus, Nature, 438 (2005), 575-576.  doi: 10.1038/438575a. [16] Y. Liu, Z. 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. [17] Q. Li, X. Guan and P. Wu, Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia, N. Engl. J. Med., 382 (2020), 1199-1207. [18] J. Ma and D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol., 68 (2006), 679-702.  doi: 10.1007/s11538-005-9047-7. [19] I. Miller, A. D. Becker, B. T. Grenfell and C. J. E. Metcalf, et al., Mapping the burden of COVID-19 in the United States, preprint. [20] H. Nishiura, S. M. Jung and N.M. Linton, The extent of transmission of novel coronavirus in Wuhan, China, J. Clin. Med., 9 (2020), 330.  doi: 10.3934/mbe.2020148. [21] C. Rothe, M. Schunk and P. Sothmann, et al., Transmission of 2019-nCoV infection from an asymptomatic contact in Germany, N. Engl. J. Med., (2020). [22] M. A. Safi, M. Imran and A. B. Gumel, Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation, Theory Biosci., 131 (2012), 19-30. [23] R. J. Shattock, M. Warren and S. McCormack, Turning the tide against HIV, Science, 333 (2011), 42-43. [24] Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.  doi: 10.1137/120876642. [25] B. Tang, X. Wang and Q. Li, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 462. [26] B. Tang, N. L. Bragazzi and Q. Li, An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov), Infectious disease modelling, 5 (2020), 248-255. [27] H. Thieme, Mathematics in Population Biology, Princeton University Press, USA, 2003. [28] 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.  doi: 10.1016/S0025-5564(02)00108-6. [29] R. Verity, L. C. Okell and I. Dorigatti, Estimates of the severity of coronavirus disease 2019: a model-based analysis, Lancet. Infect. Dis., 20 (2020), 669-677. [30] J. Wu, K. 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. [31] P. Zhou, X. Yang, X. Wang and B. Hu, et al., Discovery of a novel coronavirus associated with the recent pneumonia outbreak in humans and its potential bat origin, preprint. doi: 10.1101/2020.01.22.914952. [32] N. Zhu, D. Zhang and W. Wang, A novel coronavirus from patients with pneumonia in China, N. Engl. J. Med., 382 (2020), 727-733.  doi: 10.3934/mbe.2020148.

show all references

##### References:
 [1] V. Andreasen, The final size of an epidemic and its relation to the basic reproduction number, Bull. Math. Biol., 73 (2011), 2305-2321.  doi: 10.1007/s11538-010-9623-3. [2] J. A. Backer, D. Klinkenberg and J. Wallinga, Incubation period of 2019 novel coronavirus (2019-nCoV) infections among travellers from Wuhan, China, Eurosurveillance, 25 (2020), 2000062. [3] M. V. Barbarossa, A. D$\mathrm{\acute{e}}$nes and G. Kiss, et al., Transmission dynamics and final epidemic size of Ebola virus disease outbreaks with varying interventions, PloS. one, 10(7), (2015). [4] P. Brasil, Jr. J. P. Pereira and M. E. Moreira, Zika virus infection in pregnant women in Rio de Janeiro, N. Engl. J. Med., 375 (2016), 2321-2334.  doi: 10.1056/NEJMoa1602412. [5] F. Brauer, Early estimates of epidemic final sizes, J. Biol. Dynam., 13 (2019), 23-30. doi: 10.1080/17513758.2018.1469792. [6] F. Brauer, The final size of a serious epidemic, Bull. Math. Biol., 81 (2019), 869-877.  doi: 10.1007/s11538-018-00549-x. [7] T. Britton, F. Ball and P. Trapman, A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2, Science, 369 (2020), 846-849.  doi: 10.1126/science.abc6810. [8] S. Y. Del Valle, J. M. Hyman and N. Chitnis, Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases, Math. Bio. Eng., 10 (2013), 1475-1497.  doi: 10.3934/mbe.2013.10.1475. [9] A. D$\mathrm{\acute{e}}$nes and A. B. Gumel, Modeling the impact of quarantine during an outbreak of Ebola virus disease, Infectious Disease Modelling, 4 (2019), 12-27. [10] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324. [11] T. House, J. V. Ross and D. Sirl, How big is an outbreak likely to be? Methods for epidemic final-size calculation, Proc. R. Soc. A., 469 (2013), 20120436.  doi: 10.1098/rspa.2012.0436. [12] N. Imai, I. Dorigatti and A. Cori, et al., Report 2: Estimating the potential total number of novel Coronavirus cases in Wuhan City, China (2020)., [13] R. M. Jasmer, P. Nahid and P. C. Hopewell, Latent tuberculosis infection, N. Engl. J. Med., 347 (2002), 1860-1866. [14] D. G. Lalloo, D. Shingadia and G. Pasvol, UK malaria treatment guidelines, J. Infect., 54 (2007), 111-121. [15] E. M. Leroy, B. Kumulungui and X. Pourrut, Fruit bats as reservoirs of Ebola virus, Nature, 438 (2005), 575-576.  doi: 10.1038/438575a. [16] Y. Liu, Z. 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. [17] Q. Li, X. Guan and P. Wu, Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia, N. Engl. J. Med., 382 (2020), 1199-1207. [18] J. Ma and D. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol., 68 (2006), 679-702.  doi: 10.1007/s11538-005-9047-7. [19] I. Miller, A. D. Becker, B. T. Grenfell and C. J. E. Metcalf, et al., Mapping the burden of COVID-19 in the United States, preprint. [20] H. Nishiura, S. M. Jung and N.M. Linton, The extent of transmission of novel coronavirus in Wuhan, China, J. Clin. Med., 9 (2020), 330.  doi: 10.3934/mbe.2020148. [21] C. Rothe, M. Schunk and P. Sothmann, et al., Transmission of 2019-nCoV infection from an asymptomatic contact in Germany, N. Engl. J. Med., (2020). [22] M. A. Safi, M. Imran and A. B. Gumel, Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation, Theory Biosci., 131 (2012), 19-30. [23] R. J. Shattock, M. Warren and S. McCormack, Turning the tide against HIV, Science, 333 (2011), 42-43. [24] Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513-1532.  doi: 10.1137/120876642. [25] B. Tang, X. Wang and Q. Li, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 462. [26] B. Tang, N. L. Bragazzi and Q. Li, An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov), Infectious disease modelling, 5 (2020), 248-255. [27] H. Thieme, Mathematics in Population Biology, Princeton University Press, USA, 2003. [28] 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.  doi: 10.1016/S0025-5564(02)00108-6. [29] R. Verity, L. C. Okell and I. Dorigatti, Estimates of the severity of coronavirus disease 2019: a model-based analysis, Lancet. Infect. Dis., 20 (2020), 669-677. [30] J. Wu, K. 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. [31] P. Zhou, X. Yang, X. Wang and B. Hu, et al., Discovery of a novel coronavirus associated with the recent pneumonia outbreak in humans and its potential bat origin, preprint. doi: 10.1101/2020.01.22.914952. [32] N. Zhu, D. Zhang and W. Wang, A novel coronavirus from patients with pneumonia in China, N. Engl. J. Med., 382 (2020), 727-733.  doi: 10.3934/mbe.2020148.
Flow diagram of the model for studying COVID-19 transmission
Architecture of disease transmission flow with two age groups
The stability of the endemic equilibrium $M^\ast$ in the model (2.1). (a) The initial values are $(500, 0, 0, 1, 0, 0)$; (b) The initial values are $(5000, 0, 0, 1, 0, 0)$
The stability of the endemic equilibrium in the model (3.1). The initial values are $(500, 0, 0, 0, 0, 1, 0, 0,100, 0, 0, 0, 0, 1, 0, 0)$
The relationship between the reduction of contact rates and infection cases in Canada
The relationship between the change of contact rates and infection cases for two groups in Canada
The relationship between the reduction of contact rates and infection cases in NL
Parameter Description and Estimation
 Parameter Description Values(per day) Source $\Pi$ Recruitment rate 136 [22] $\mu$ Natural death rate 0.0078 United Nations $\beta_1$ Effective contact rate with $A$ 0.491 [9] $\beta_2$ Effective contact rate with $I$ 0.391 assumed $\beta_3$ Effective contact rate with $Q$ 0.291 assumed $\alpha_{1}$ Progression rate from $E$ to $A$ 0.01880857 [25] $\alpha_{2}$ Progression rate from $E$ to $I$ 0.156986 [22] $\sigma$ Symptom progression rate($A$ to $I$) 0.001 assumed $q_{1}$ Quarantine rate for $E$ 0.1 [22] $q_{2}$ Quarantine rate for $A$ 0.11 assumed $q_{3}$ Quarantine rate for $I$ 0.12 assumed $\gamma_{1}$ Recovery rate for $A$ 0.13978 [25] $\gamma_{2}$ Recovery rate for $I$ 0.03521 [22] $\gamma_{3}$ Recovery rate for $Q$ 0.042553 [22] $\theta_{1}$ Disease-induced death rate for $A$ 0.001 assumed $\theta_{2}$ Disease-induced death rate for $I$ 0.04227 [22]
 Parameter Description Values(per day) Source $\Pi$ Recruitment rate 136 [22] $\mu$ Natural death rate 0.0078 United Nations $\beta_1$ Effective contact rate with $A$ 0.491 [9] $\beta_2$ Effective contact rate with $I$ 0.391 assumed $\beta_3$ Effective contact rate with $Q$ 0.291 assumed $\alpha_{1}$ Progression rate from $E$ to $A$ 0.01880857 [25] $\alpha_{2}$ Progression rate from $E$ to $I$ 0.156986 [22] $\sigma$ Symptom progression rate($A$ to $I$) 0.001 assumed $q_{1}$ Quarantine rate for $E$ 0.1 [22] $q_{2}$ Quarantine rate for $A$ 0.11 assumed $q_{3}$ Quarantine rate for $I$ 0.12 assumed $\gamma_{1}$ Recovery rate for $A$ 0.13978 [25] $\gamma_{2}$ Recovery rate for $I$ 0.03521 [22] $\gamma_{3}$ Recovery rate for $Q$ 0.042553 [22] $\theta_{1}$ Disease-induced death rate for $A$ 0.001 assumed $\theta_{2}$ Disease-induced death rate for $I$ 0.04227 [22]
Variable descriptions (i = 1, 2)
 Variable Description $S_i(t)$ Susceptible population $E_i(t)$ Exposed population $A_i(t)$ Infectious population without symptoms $I^1_i(t)$ Infectious population with mild symptoms $I^2_i(t)$ Patient in hospital $I^3_i(t)$ Patient in intensive care $Q_i(t)$ Quarantined population $R_i(t)$ Recovered population
 Variable Description $S_i(t)$ Susceptible population $E_i(t)$ Exposed population $A_i(t)$ Infectious population without symptoms $I^1_i(t)$ Infectious population with mild symptoms $I^2_i(t)$ Patient in hospital $I^3_i(t)$ Patient in intensive care $Q_i(t)$ Quarantined population $R_i(t)$ Recovered population
Parameter estimates (i = 1, 2)
 Parameter Description Values(per day) Source $\beta_{ij}$ Effective contact rate in each group 0.081-0.591 [9] $\alpha^1_i$ Progression rate from $E_i$ to $A_i$ 0.01881, 0.02881 [25] $\alpha^2_i$ Progression rate from $E_i$ to $I^1_i$ 0.15699, 0.16699 [22] $q^1_{i}$ Quarantine rate for $A_i$ 0.43, 0.1 [22] $q^2_{i}$ Quarantine rate for $E_i$ 0.45, 0.12 assumed $q^3_{i}$ Quarantine rate for $I^1_i$ 0.52, 0.13 assumed $\gamma^1_{i}$ Recovery rate for $A_i$ 0.13978, 0.12978 [25] $\gamma^2_{i}$ Recovery rate for $Q_i$ 0.1, 0.09 assumed $\gamma^3_{i}$ Recovery rate for $I^1_i$ 0.08, 0.06 assumed $\gamma^4_{i}$ Recovered rate for $I^2_i$ 0.06, 0.04 assumed $\gamma^5_{i}$ Recovered rate for $I^3_i$ 0.04, 0.01 assumed $s^a_i$ Symptom progression rate from $A_i$ to $I^1_i$ 0.05, 0.06 assumed $s^1_i$ Symptom progression rate from $I^1_i$ to $I^2_i$ 0.06, 0.08 assumed $s^2_i$ Symptom progression rate from $I^2_i$ to $I^3_i$ 0.07, 0.09 assumed $d^1_{i}$ Disease-induced death rate for $A_i$ 0.02, 0.03 assumed $d^2_{i}$ Disease-induced death rate for $I^1_i$ 0.04227, 0.05227 [22] $d^3_{i}$ Disease-induced death rate for $I^2_i$ 0.05227, 0.08227 assumed $d^4_{i}$ Disease-induced death rate for $I^3_i$ 0.06227, 0.1 assumed
 Parameter Description Values(per day) Source $\beta_{ij}$ Effective contact rate in each group 0.081-0.591 [9] $\alpha^1_i$ Progression rate from $E_i$ to $A_i$ 0.01881, 0.02881 [25] $\alpha^2_i$ Progression rate from $E_i$ to $I^1_i$ 0.15699, 0.16699 [22] $q^1_{i}$ Quarantine rate for $A_i$ 0.43, 0.1 [22] $q^2_{i}$ Quarantine rate for $E_i$ 0.45, 0.12 assumed $q^3_{i}$ Quarantine rate for $I^1_i$ 0.52, 0.13 assumed $\gamma^1_{i}$ Recovery rate for $A_i$ 0.13978, 0.12978 [25] $\gamma^2_{i}$ Recovery rate for $Q_i$ 0.1, 0.09 assumed $\gamma^3_{i}$ Recovery rate for $I^1_i$ 0.08, 0.06 assumed $\gamma^4_{i}$ Recovered rate for $I^2_i$ 0.06, 0.04 assumed $\gamma^5_{i}$ Recovered rate for $I^3_i$ 0.04, 0.01 assumed $s^a_i$ Symptom progression rate from $A_i$ to $I^1_i$ 0.05, 0.06 assumed $s^1_i$ Symptom progression rate from $I^1_i$ to $I^2_i$ 0.06, 0.08 assumed $s^2_i$ Symptom progression rate from $I^2_i$ to $I^3_i$ 0.07, 0.09 assumed $d^1_{i}$ Disease-induced death rate for $A_i$ 0.02, 0.03 assumed $d^2_{i}$ Disease-induced death rate for $I^1_i$ 0.04227, 0.05227 [22] $d^3_{i}$ Disease-induced death rate for $I^2_i$ 0.05227, 0.08227 assumed $d^4_{i}$ Disease-induced death rate for $I^3_i$ 0.06227, 0.1 assumed
 % of contact rate ($\beta_i$) $S_\infty$ Disease Cases ($N_1$-$S_\infty$) 1 7911534 29678466 0.8 13889287 23700713 0.6 26530691 11059309 0.55 31820452 5769548 0.5 37585587 4413 0.4 37589673 327 0.2 37589851 149
 % of contact rate ($\beta_i$) $S_\infty$ Disease Cases ($N_1$-$S_\infty$) 1 7911534 29678466 0.8 13889287 23700713 0.6 26530691 11059309 0.55 31820452 5769548 0.5 37585587 4413 0.4 37589673 327 0.2 37589851 149
Cases over time in Canada for two groups
 % $\beta_{ij}$ $S_1(\infty)$ $S_2(\infty)$ Disease cases (I) $N_1^0-S_1(\infty)$ % $1-\frac{S_1(\infty)}{N_1^0}$ Disease cases (II) $N_2^0-S_2(\infty)$ % $1-\frac{S_2(\infty)}{N_2^0}$ Total cases 1 9811036 1091182 21313484 0.68 5374298 0.83 26687782 0.8 16552490 1961078 14572030 0.47 4504402 0.7 19076432 0.6 30413213 3831763 711307 0.02 2633717 0.41 3345024 0.4 31124381 6465416 139 $4.5\times10^{-6}$ 64 $9.9\times10^{-6}$ 203 0.2 31124437 6465461 83 $2.7\times10^{-6}$ 19 $2.9\times10^{-6}$ 102
 % $\beta_{ij}$ $S_1(\infty)$ $S_2(\infty)$ Disease cases (I) $N_1^0-S_1(\infty)$ % $1-\frac{S_1(\infty)}{N_1^0}$ Disease cases (II) $N_2^0-S_2(\infty)$ % $1-\frac{S_2(\infty)}{N_2^0}$ Total cases 1 9811036 1091182 21313484 0.68 5374298 0.83 26687782 0.8 16552490 1961078 14572030 0.47 4504402 0.7 19076432 0.6 30413213 3831763 711307 0.02 2633717 0.41 3345024 0.4 31124381 6465416 139 $4.5\times10^{-6}$ 64 $9.9\times10^{-6}$ 203 0.2 31124437 6465461 83 $2.7\times10^{-6}$ 19 $2.9\times10^{-6}$ 102
Cases over time in NL
 % of contact rate ($\beta_i$) $S_\infty$ Disease Cases ($N_1$-$S_\infty$) 1 109772 411770 0.8 192707 328835 0.6 368064 153478 0.5 520569 973 0.4 521464 78 0.2 521506 36
 % of contact rate ($\beta_i$) $S_\infty$ Disease Cases ($N_1$-$S_\infty$) 1 109772 411770 0.8 192707 328835 0.6 368064 153478 0.5 520569 973 0.4 521464 78 0.2 521506 36
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