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doi: 10.3934/dcdsb.2022067
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Numerical threshold of linearly implicit Euler method for nonlinear infection-age SIR models

1. 

Automation, Southeast University, Nanjing, 210096, China

2. 

School of Mathematics and Statistic Science, Ludong University, Yantai, 264025, China

3. 

School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

4. 

School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China

* Corresponding author: yangzhan_wen@hit.edu.cn

Received  March 2021 Revised  February 2022 Early access March 2022

Fund Project: The corresponding author is supported by NSFC grant 11871179 and 11771128

In this paper, we consider a numerical threshold of a linearly implicit Euler method for a nonlinear infection-age SIR model. It is shown that the method shares the equilibria and basic reproduction number $ R_0 $ of age-independent SIR models for any stepsize. Namely, the disease-free equilibrium is globally stable for numerical processes when $ R_0<1 $ and the underlying endemic equilibrium is globally stable for numerical processes when $ R_0>1 $. A natural extension to nonlinear infection-age models is presented with an initial mortality rate and the numerical thresholds, i.e., numerical basic reproduction numbers $ R^h $, are presented according to the infinite Leslie matrix. Although the numerical basic reproduction numbers $ R^h $ are not quadrature approximations to the exact threshold $ R_0 $, the disease-free equilibrium is locally stable for numerical processes whenever $ R^h<1 $. Moreover, a unique numerical endemic equilibrium exists for $ R^h>1 $, which is locally stable for numerical processes. It is much more important that both the numerical thresholds and numerical endemic equilibria converge to the exact ones with accuracy of order 1. Therefore, the local dynamical behaviors of nonlinear infection-age models are visually displayed by the numerical processes. Finally, numerical applications to the influenza models are shown to illustrate our results.

Citation: Huizi Yang, Zhanwen Yang, Shengqiang Liu. Numerical threshold of linearly implicit Euler method for nonlinear infection-age SIR models. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022067
References:
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D. Breda, Methods for numerical computation of characteristic roots for delay differential equations: Experimental comparison, Sci. Math. Jpn., 58 (2003), 377-388. 

[2]

D. Breda, Solution operator approximation for characteristic roots of delay differential equations, Appl. Numer. Math., 56 (2006), 305-317.  doi: 10.1016/j.apnum.2005.04.010.

[3]

D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15, (2016), 1–23. doi: 10.1137/15M1040931.

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D. Breda, F. Florian, J. Ripoll and R. Vermiglio, Efficient numerical computation of the basic reproduction number for structured populations, J. Comput. Appl. Math., 384 (2021), Paper No. 113165, 15 pp. doi: 10.1016/j.cam.2020.113165.

[5]

D. Breda, T. Kuniya, J. Ripoll and R. Vermiglio, Collocation of next-generation operators for computing the basic reproduction number of structured populations, J. Sci. Comput., 85 (2020), Paper No. 40, 33 pp. doi: 10.1007/s10915-020-01339-1.

[6]

D. BredaS. Maset and R. Vermiglio, Computing the characteristic roots for delay differential equations, IMA J. Numer. Anal, 24 (2004), 1-19.  doi: 10.1093/imanum/24.1.1.

[7]

D. BredaS. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495.  doi: 10.1137/030601600.

[8]

D. BredaS. Maset and R. Vermiglio, Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics, Discrete Contin. Dyn. Syst., 32 (2012), 2675-2699.  doi: 10.3934/dcds.2012.32.2675.

[9]

D. BredaR. Vermiglio and S. Maset, Computing the eigenvalues of Gurtin-MacCamy models with diffusion, IMA J. Numer. Anal., 32 (2012), 1030-1050.  doi: 10.1093/imanum/drr004.

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L. Demetrius, On an infinite population matrix, Math. Biosci., 13 (1972), 133-137.  doi: 10.1016/0025-5564(72)90029-6.

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O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction number $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[12]

F. Gosselin and J. D. Lebreton, Asymptotic properties of infinite Leslie matrices, J. Theoret. Biol., 256 (2009), 157-163.  doi: 10.1016/j.jtbi.2008.09.018.

[13]

W. J. GuoM. YeX. N. LiA. Meyer-Baese and Q. M. Zhang, A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon, Math. Biosci. Eng., 16 (2019), 4107-4121.  doi: 10.3934/mbe.2019204.

[14]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.  doi: 10.1007/BF00250793.

[15]

M. IannelliF. A. Milner and A. Pugliese, Analytical and numerical results for the age-structured SIS epidemic model with mixed inter-intracohort transmission, SIAM J. Math. Anal., 23 (1992), 662-688.  doi: 10.1137/0523034.

[16]

H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.  doi: 10.1007/BF00178326.

[17]

T. Kuniya, Numerical approximation of the basic reproduction number for a class of age-structured epidemic models, Appl. Math. Lett., 73 (2017), 106-112.  doi: 10.1016/j.aml.2017.04.031.

[18]

P. MagalC. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.

[19]

P. Magal and S. G. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer, 2018. doi: 10.1007/978-3-030-01506-0.

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P. W. Nelson, M. A. Gilchrist and D. Coombs, et al., An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267.

[21]

F. ScarabelD. BredaO. DiekmannM. Gyllenberg and R. Vermiglio, Numerical bifurcation analysis of physiologically structured population models via pseudospectral approximation, Vietnam J. Math., 49 (2021), 37-67.  doi: 10.1007/s10013-020-00421-3.

[22]

F. Scarabel, O. Diekmann and R. Vermiglio, Numerical bifurcation analysis of renewal equations via pseudospectral approximation, J. Comput. Appl. Math., 397 (2021), Paper No. 113611, 21 pp. doi: 10.1016/j.cam.2021.113611.

[23]

F. R. Sharpe and A. J. Lotka, A problem in age-distribution, Philos. Mag. Ser., 21 (1991), 435-438.  doi: 10.1007/978-3-642-81046-6_13.

[24]

H. R. Thieme and C. Castillo-Chavez, On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic, Mathematical and Statistical Approaches to AIDS Epidemiology, 83 (1989), 157-176.  doi: 10.1007/978-3-642-93454-4_7.

[25]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM. J. Appl. Math., 53 (1993), 1447-1479.  doi: 10.1137/0153068.

[26]

X. H. TianR. Xu and J. Z. Lin, Mathematical analysis of an age-structured HIV-1 infection model with CTL immune response, Math. Biosci. Eng., 16 (2019), 7850-7882.  doi: 10.3934/mbe.2019395.

[27]

J. L. WangM. Guo and S. Q. Liu, SVIR epidemic model with age-structure in susceptibility, vaccination effects and relapse, IMA J. Appl. Math., 82 (2017), 945-970.  doi: 10.1093/imamat/hxx020.

[28]

D. X. Yan and X. L. Fu, Analysis of an age-structured HIV infection model with logistic target-cell growth and antiretroviral therapy, IMA J. Appl. Math., 83 (2018), 1037-1065.  doi: 10.1093/imamat/hxy034.

[29]

Q. Z. Yang and Y. N. Yang, Further results for Perron-Frobenius theorem 275 for nonnegative tensors II, SIAM J. Matrix Anal. Appl., 32 (2011), 1236-1250.  doi: 10.1137/100813671.

[30]

Q. Z. Yang and Y. N. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.  doi: 10.1137/090778766.

[31]

Z. W. YangT. Q. Zuo and Z. J. Chen, Numerical analysis of linearly implicit Euler-Riemann method for nonlinear Gurtin-MacCamy model, Appl. Numer. Math., 163 (2021), 147-166.  doi: 10.1016/j.apnum.2020.12.018.

show all references

References:
[1]

D. Breda, Methods for numerical computation of characteristic roots for delay differential equations: Experimental comparison, Sci. Math. Jpn., 58 (2003), 377-388. 

[2]

D. Breda, Solution operator approximation for characteristic roots of delay differential equations, Appl. Numer. Math., 56 (2006), 305-317.  doi: 10.1016/j.apnum.2005.04.010.

[3]

D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15, (2016), 1–23. doi: 10.1137/15M1040931.

[4]

D. Breda, F. Florian, J. Ripoll and R. Vermiglio, Efficient numerical computation of the basic reproduction number for structured populations, J. Comput. Appl. Math., 384 (2021), Paper No. 113165, 15 pp. doi: 10.1016/j.cam.2020.113165.

[5]

D. Breda, T. Kuniya, J. Ripoll and R. Vermiglio, Collocation of next-generation operators for computing the basic reproduction number of structured populations, J. Sci. Comput., 85 (2020), Paper No. 40, 33 pp. doi: 10.1007/s10915-020-01339-1.

[6]

D. BredaS. Maset and R. Vermiglio, Computing the characteristic roots for delay differential equations, IMA J. Numer. Anal, 24 (2004), 1-19.  doi: 10.1093/imanum/24.1.1.

[7]

D. BredaS. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495.  doi: 10.1137/030601600.

[8]

D. BredaS. Maset and R. Vermiglio, Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics, Discrete Contin. Dyn. Syst., 32 (2012), 2675-2699.  doi: 10.3934/dcds.2012.32.2675.

[9]

D. BredaR. Vermiglio and S. Maset, Computing the eigenvalues of Gurtin-MacCamy models with diffusion, IMA J. Numer. Anal., 32 (2012), 1030-1050.  doi: 10.1093/imanum/drr004.

[10]

L. Demetrius, On an infinite population matrix, Math. Biosci., 13 (1972), 133-137.  doi: 10.1016/0025-5564(72)90029-6.

[11]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction number $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[12]

F. Gosselin and J. D. Lebreton, Asymptotic properties of infinite Leslie matrices, J. Theoret. Biol., 256 (2009), 157-163.  doi: 10.1016/j.jtbi.2008.09.018.

[13]

W. J. GuoM. YeX. N. LiA. Meyer-Baese and Q. M. Zhang, A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon, Math. Biosci. Eng., 16 (2019), 4107-4121.  doi: 10.3934/mbe.2019204.

[14]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.  doi: 10.1007/BF00250793.

[15]

M. IannelliF. A. Milner and A. Pugliese, Analytical and numerical results for the age-structured SIS epidemic model with mixed inter-intracohort transmission, SIAM J. Math. Anal., 23 (1992), 662-688.  doi: 10.1137/0523034.

[16]

H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.  doi: 10.1007/BF00178326.

[17]

T. Kuniya, Numerical approximation of the basic reproduction number for a class of age-structured epidemic models, Appl. Math. Lett., 73 (2017), 106-112.  doi: 10.1016/j.aml.2017.04.031.

[18]

P. MagalC. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122.

[19]

P. Magal and S. G. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer, 2018. doi: 10.1007/978-3-030-01506-0.

[20]

P. W. Nelson, M. A. Gilchrist and D. Coombs, et al., An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267.

[21]

F. ScarabelD. BredaO. DiekmannM. Gyllenberg and R. Vermiglio, Numerical bifurcation analysis of physiologically structured population models via pseudospectral approximation, Vietnam J. Math., 49 (2021), 37-67.  doi: 10.1007/s10013-020-00421-3.

[22]

F. Scarabel, O. Diekmann and R. Vermiglio, Numerical bifurcation analysis of renewal equations via pseudospectral approximation, J. Comput. Appl. Math., 397 (2021), Paper No. 113611, 21 pp. doi: 10.1016/j.cam.2021.113611.

[23]

F. R. Sharpe and A. J. Lotka, A problem in age-distribution, Philos. Mag. Ser., 21 (1991), 435-438.  doi: 10.1007/978-3-642-81046-6_13.

[24]

H. R. Thieme and C. Castillo-Chavez, On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic, Mathematical and Statistical Approaches to AIDS Epidemiology, 83 (1989), 157-176.  doi: 10.1007/978-3-642-93454-4_7.

[25]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM. J. Appl. Math., 53 (1993), 1447-1479.  doi: 10.1137/0153068.

[26]

X. H. TianR. Xu and J. Z. Lin, Mathematical analysis of an age-structured HIV-1 infection model with CTL immune response, Math. Biosci. Eng., 16 (2019), 7850-7882.  doi: 10.3934/mbe.2019395.

[27]

J. L. WangM. Guo and S. Q. Liu, SVIR epidemic model with age-structure in susceptibility, vaccination effects and relapse, IMA J. Appl. Math., 82 (2017), 945-970.  doi: 10.1093/imamat/hxx020.

[28]

D. X. Yan and X. L. Fu, Analysis of an age-structured HIV infection model with logistic target-cell growth and antiretroviral therapy, IMA J. Appl. Math., 83 (2018), 1037-1065.  doi: 10.1093/imamat/hxy034.

[29]

Q. Z. Yang and Y. N. Yang, Further results for Perron-Frobenius theorem 275 for nonnegative tensors II, SIAM J. Matrix Anal. Appl., 32 (2011), 1236-1250.  doi: 10.1137/100813671.

[30]

Q. Z. Yang and Y. N. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.  doi: 10.1137/090778766.

[31]

Z. W. YangT. Q. Zuo and Z. J. Chen, Numerical analysis of linearly implicit Euler-Riemann method for nonlinear Gurtin-MacCamy model, Appl. Numer. Math., 163 (2021), 147-166.  doi: 10.1016/j.apnum.2020.12.018.

Figure 1.  The numerical susceptible and infectious individuals for $ A = 365 $, $ \mu_S = \frac{1}{365} $ and $ i(a, 0) = 50(a+2)e^{-0.4(a+2)} $
Figure 2.  The numerical distributions for $ A = 365 $, $ \mu_S = \frac{1}{365} $ and $ i(a, 0) = 50(a+2)e^{-0.4(a+2)} $ with different initial value $ S(0) $
Figure 3.  Numerical basic reproduction numbers $ R^h $ against $ (\sigma_1, \sigma_2) $ and $ (a_1, a_2) $ for $ h = 0.02 $
Figure 4.  Numerical susceptible individuals and total infections for $ a_1 = 0 $ and $ a_2 = 7 $
Figure 5.  Numerical distributions for $ h = 0.1 $, $ a_1 = 0 $ and $ a_2 = 7 $
Figure 6.  Numerical susceptible individuals and total infections for $ a_1 = 0 $ and $ a_2 = 21 $
Figure 7.  Numerical distributions for $ h = 0.1 $, $ a_1 = 0 $ and $ a_2 = 21 $
Figure 8.  The occurrence and outbreaks of the disease for $ R_0>1 $
Figure 9.  The treatment and isolation after a while
Table 1.  Convergence orders of numerical susceptible individual and total infections with related errors at time level $ T = 1 $
$ h $ $ r_S^h $ $ p_S $ $ r_I^h $ $ p_I $
0.1 $ 2.7619E-2 $ $ 2.1627E-3 $
0.05 $ 1.4217E-2 $ 0.9580 $ 1.1056E-3 $ 0.9680
0.025 $ 7.2120E-3 $ 0.9792 $ 5.5920E-4 $ 0.9835
0.0125 $ 3.6321E-3 $ 0.9896 $ 2.8123E-4 $ 0.9916
0.00625 $ 1.8226E-3 $ 0.9948 $ 1.4103E-4 $ 0.9958
$ h $ $ r_S^h $ $ p_S $ $ r_I^h $ $ p_I $
0.1 $ 2.7619E-2 $ $ 2.1627E-3 $
0.05 $ 1.4217E-2 $ 0.9580 $ 1.1056E-3 $ 0.9680
0.025 $ 7.2120E-3 $ 0.9792 $ 5.5920E-4 $ 0.9835
0.0125 $ 3.6321E-3 $ 0.9896 $ 2.8123E-4 $ 0.9916
0.00625 $ 1.8226E-3 $ 0.9948 $ 1.4103E-4 $ 0.9958
Table 2.  Convergence orders of numerical basic reproduction numbers with related errors
$ \sigma_1 = 0.66667, \sigma_2 = 0.6 $ $ \sigma_1 = 0.1111, \sigma_2 = 2.8 $ $ \sigma_1 = 0.2, \sigma_2 = 1 $
$ h $ $ r_R^h $ $ p_R $ $ r_R^h $ $ p_R $ $ r_R^h $ $ p_R $
0.1 $ 2.3927E-2 $ $ 7.3455E-3 $ $ 1.7730E-2 $
0.05 $ 1.2390E-2 $ 0.9494 $ 3.8549E-3 $ 0.9302 $ 9.1923E-3 $ 0.9477
0.025 $ 6.3070E-3 $ 0.9742 $ 1.9703E-3 $ 0.9683 $ 4.6817E-3 $ 0.9734
0.0125 $ 3.1821E-3 $ 0.9870 $ 9.9581E-4 $ 0.9845 $ 2.3627E-3 $ 0.9866
0.00625 $ 1.5983E-3 $ 0.9934 $ 5.0058E-4 $ 0.9923 $ 1.1869E-3 $ 0.9933
$ \sigma_1 = 0.66667, \sigma_2 = 0.6 $ $ \sigma_1 = 0.1111, \sigma_2 = 2.8 $ $ \sigma_1 = 0.2, \sigma_2 = 1 $
$ h $ $ r_R^h $ $ p_R $ $ r_R^h $ $ p_R $ $ r_R^h $ $ p_R $
0.1 $ 2.3927E-2 $ $ 7.3455E-3 $ $ 1.7730E-2 $
0.05 $ 1.2390E-2 $ 0.9494 $ 3.8549E-3 $ 0.9302 $ 9.1923E-3 $ 0.9477
0.025 $ 6.3070E-3 $ 0.9742 $ 1.9703E-3 $ 0.9683 $ 4.6817E-3 $ 0.9734
0.0125 $ 3.1821E-3 $ 0.9870 $ 9.9581E-4 $ 0.9845 $ 2.3627E-3 $ 0.9866
0.00625 $ 1.5983E-3 $ 0.9934 $ 5.0058E-4 $ 0.9923 $ 1.1869E-3 $ 0.9933
Table 3.  Convergence orders of numerical total infection numbers with related errors
$ \sigma_1 = 0.66667, \sigma_2 = 0.6 $ $ \sigma_1 = 0.1111, \sigma_2 = 2.8 $ $ \sigma_1 = 0.2, \sigma_2 = 1 $
$ h $ $ r_{I^*}^h $ $ p_{I^*} $ $ r_{I^*}^h $ $ p_{I^*} $ $ r_{I^*}^h $ $ p_{I^*} $
0.1 $ 5.4581E-3 $ $ 1.0007E-2 $ $ 5.4661E-3 $
0.05 $ 2.7541E-3 $ 0.9868 $ 5.1620E-3 $ 0.9550 $ 2.7569E-3 $ 0.9975
0.025 $ 1.3834E-3 $ 0.9934 $ 2.6234E-3 $ 0.9765 $ 1.3845E-3 $ 0.9937
0.0125 $ 6.9327E-4 $ 0.9967 $ 1.3225E-3 $ 0.9882 $ 6.9378E-4 $ 0.9968
0.00625 $ 3.4703E-4 $ 0.9983 $ 6.6398E-4 $ 0.9941 $ 3.4727E-4 $ 0.9984
$ \sigma_1 = 0.66667, \sigma_2 = 0.6 $ $ \sigma_1 = 0.1111, \sigma_2 = 2.8 $ $ \sigma_1 = 0.2, \sigma_2 = 1 $
$ h $ $ r_{I^*}^h $ $ p_{I^*} $ $ r_{I^*}^h $ $ p_{I^*} $ $ r_{I^*}^h $ $ p_{I^*} $
0.1 $ 5.4581E-3 $ $ 1.0007E-2 $ $ 5.4661E-3 $
0.05 $ 2.7541E-3 $ 0.9868 $ 5.1620E-3 $ 0.9550 $ 2.7569E-3 $ 0.9975
0.025 $ 1.3834E-3 $ 0.9934 $ 2.6234E-3 $ 0.9765 $ 1.3845E-3 $ 0.9937
0.0125 $ 6.9327E-4 $ 0.9967 $ 1.3225E-3 $ 0.9882 $ 6.9378E-4 $ 0.9968
0.00625 $ 3.4703E-4 $ 0.9983 $ 6.6398E-4 $ 0.9941 $ 3.4727E-4 $ 0.9984
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