Article Contents

2023, Volume 28, Issue 1: 70-92. 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
^* Corresponding author: yangzhan_wen@hit.edu.cn

Received: March 2021

Revised: February 2022

Early access: April 2022

Published: January 2023

The corresponding author is supported by NSFC grant 11871179 and 11771128

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Abstract

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.

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Mathematics Subject Classification: Primary: 65M20, 65P40; Secondary: 92B05.

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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)} $

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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) $

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Figure 3. Numerical basic reproduction numbers $ R^h $ against $ (\sigma_1, \sigma_2) $ and $ (a_1, a_2) $ for $ h = 0.02 $

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Figure 4. Numerical susceptible individuals and total infections for $ a_1 = 0 $ and $ a_2 = 7 $

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Figure 5. Numerical distributions for $ h = 0.1 $, $ a_1 = 0 $ and $ a_2 = 7 $

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Figure 6. Numerical susceptible individuals and total infections for $ a_1 = 0 $ and $ a_2 = 21 $

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Figure 7. Numerical distributions for $ h = 0.1 $, $ a_1 = 0 $ and $ a_2 = 21 $

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Figure 8. The occurrence and outbreaks of the disease for $ R_0>1 $

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Figure 9. The treatment and isolation after a while

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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

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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

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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

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