Positivity-Preserving Numerical Method for a Stochastic Multi-Group SIR Epidemic Model

Han Ma; Qimin Zhang; Xinzhong Xu

doi:10.1515/cmam-2022-0143

Article

Positivity-Preserving Numerical Method for a Stochastic Multi-Group SIR Epidemic Model

Han Ma , Qimin Zhang and Xinzhong Xu

Published/Copyright: December 7, 2022

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From the journal Computational Methods in Applied Mathematics Volume 23 Issue 3

Abstract

The stochastic multi-group susceptible–infected–recovered (SIR) epidemic model is nonlinear, and so analytical solutions are generally difficult to obtain. Hence, it is often necessary to find numerical solutions, but most existing numerical methods fail to preserve the nonnegativity or positivity of solutions. Therefore, an appropriate numerical method for studying the dynamic behavior of epidemic diseases through SIR models is urgently required. In this paper, based on the Euler–Maruyama scheme and a logarithmic transformation, we propose a novel explicit positivity-preserving numerical scheme for a stochastic multi-group SIR epidemic model whose coefficients violate the global monotonicity condition. This scheme not only results in numerical solutions that preserve the domain of the stochastic multi-group SIR epidemic model, but also achieves the “ $order - \frac{1}{2}$ $order - 1 2$ ” strong convergence rate. Taking a two-group SIR epidemic model as an example, some numerical simulations are performed to illustrate the performance of the proposed scheme.

Keywords: Stochastic Multi-Group SIR Epidemic Model; Positivity-Preserving; Logarithmic Truncated Euler–Maruyama Method; Strong Convergence

MSC 2010: 60H10; 92D30; 60J27

Funding statement: The authors thank the editor and referees for their careful reading and valuable comments. The research was supported in part by National Natural Science Foundation of China (No. 12161068) and Ningxia Key R&D Program Key Projects (No. 2021BEG03012).

A Proof of Theorem 3.1

For any given initial value $( I k ⁢ 0 , R k ⁢ 0 ) ∈ ℝ + 2 ⁢ n$ , $k = 1 , 2 , … , n$ , it is easy to know that there is a unique local solution $( I k ⁢ ( t ) , R k ⁢ ( t ) )$ on $t ∈ [ 0 , τ e )$ because the coefficients of the model (2.1) are locally Lipschitz continuous, where $τ e$ is the explosion time (see [3, 9]). In order to demonstrate the local solution is global, we have to prove that $τ e = ∞$ a.s. Let $m 0$ be positive number sufficiently large such that all components of $Z k ⁢ 0$ are included in the interval $[ 1 m 0 , m 0 ]$ . For each integer $m ≥ m 0$ , there are definition of the stopping time

$τ m = inf ⁡ { t ∈ [ 0 , τ e ) : min ⁡ { I k ⁢ ( t ) , R k ⁢ ( t ) : k = 1 , 2 , … , n } ≤ 1 m ⁢ or ⁢ max ⁡ { I k ⁢ ( t ) , R k ⁢ ( t ) : k = 1 , 2 , … , n } ≥ m } ,$

we set $inf ⁡ ∅ = ∞$ (the $∅$ denotes the empty set). Obviously, the $τ m$ is increasing as $m → ∞$ . Set

$τ ∞ = lim m → ∞ ⁡ τ m when τ ∞ ≤ τ e a.s.$

If we can prove that $τ ∞ = ∞$ a.s., then the results of $τ e = ∞$ and $( I k ⁢ ( t ) , R k ⁢ ( t ) ) ∈ ℝ + 2 ⁢ n$ a.s. can be obtained for all $t ≥ 0$ . Therefore, on analysis of the preceding, we have to illustrate $τ ∞ = ∞$ a.s. to complete the proof. If this statement is false, then there exists a pair of constants $T > 0$ and $ϵ ∈ ( 0 , 1 )$ such that

$ℙ { τ ∞ ≤ T } > ϵ .$

Thus, there exists an integer $m 1 ≥ m 0$ such that

(A.1)

$ℙ { τ ∞ ≤ T } ≥ ϵ for all m ≥ m 1 .$

Here, we define a $C 2$ -function $V : ℝ + 2 ⁢ n → ℝ +$ by

$V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) ) = ∑ k = 1 n [ I k ⁢ ( t ) + 1 - log ⁡ ( I k ⁢ ( t ) ) ] + [ R k ⁢ ( t ) + 1 - log ⁡ ( R k ⁢ ( t ) ) ] .$

The function V is nonnegative, which is easy to see from $u + 1 - log ⁡ ( u ) ≥ 0$ for all $u > 0$ . Let $ℒ$ be the generating operator of system (2.1), it is easy to see that

$ℒ V = ∑ k = 1 n [ ( 1 - 1 I k ⁢ ( t ) ) ( λ k ( t ) + β k ⁢ k ( N k - R k ( t ) ) I k ( t ) - ( d k I ( t ) + ε k + γ k ) I k ( t ) - β k ⁢ k I k 2 ( t ) )$

$+ 1 2 σ k 2 ( N k - R k ( t ) - I k ( t ) ) 2 ] + ∑ k = 1 n [ ( 1 - 1 R k ⁢ ( t ) ) ( γ k I k ( t ) - d k R R k ( t ) ) ]$

$= ∑ k = 1 n [ λ k ( t ) + β k ⁢ k ( N k - R k ( t ) ) I k ( t ) - λ k ⁢ ( t ) I k ⁢ ( t ) - ( d k I + ε k + γ k ) I k ( t ) - β k ⁢ k I k 2 ( t ) - β k ⁢ k ( N k - R k ( t ) )$

$+ ( d k R + d k I + ε k + γ k ) + β k ⁢ k I k ( t ) + γ k I k ( t ) - d k R R k ( t ) - γ k ⁢ I k ⁢ ( t ) R k ⁢ ( t ) + 1 2 σ k 2 ( N k - R k ( t ) - I k ( t ) ) 2 ]$

$≤ ∑ k = 1 n [ λ k ⁢ ( t ) + ( β k ⁢ k ⁢ N k + β k ⁢ k + γ k ) ⁢ I k ⁢ ( t ) + ( d k R + d k I + ε k + γ k ) + 1 2 ⁢ σ k 2 ⁢ N k 2 ]$

$≤ ∑ k = 1 n [ λ ¯ k + ( β k ⁢ k ⁢ N k + β k ⁢ k + γ k ) ⁢ I k ⁢ ( t ) + ( d k R + d k I + ε k + γ k ) + 1 2 ⁢ σ k 2 ⁢ N k 2 ] .$

Now from Lemma 1 and definition of V,

$I k ⁢ ( t ) ≤ 2 ⁢ V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) ) - ( 4 - 2 ⁢ log ⁡ 2 ) , k = 1 , 2 , … , n .$

Hence we get

$d ⁢ V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) )$

$= ℒ ⁢ V ⁢ d ⁢ t + [ ∑ k = 1 n ( σ k ⁢ ( N k - R k ⁢ ( t ) - I k ⁢ ( t ) ) ⁢ ( I k ⁢ ( t ) - 1 ) ) ] ⁢ d ⁢ B k ⁢ ( t )$

$≤ [ ∑ k = 1 n ( λ ¯ k + 2 ⁢ ( β k ⁢ k ⁢ N k + β k ⁢ k + γ k ) ⁢ V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) ) + ( d k R + d k I + ε k + γ k ) + 1 2 ⁢ σ k 2 ⁢ N k 2 ) ] ⁢ d ⁢ t$

$+ [ ∑ k = 1 n ( σ k ⁢ ( N k - R k ⁢ ( t ) - I k ⁢ ( t ) ) ⁢ ( I k ⁢ ( t ) - 1 ) ) ] ⁢ d ⁢ B k ⁢ ( t )$

$≤ [ C ⁢ ( 1 + V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) ) ) ] ⁢ d ⁢ t + [ ∑ k = 1 n ( σ k ⁢ ( N k - R k ⁢ ( t ) - I k ⁢ ( t ) ) ⁢ ( I k ⁢ ( t ) - 1 ) ) ] ⁢ d ⁢ B k ⁢ ( t ) ,$

where

$C = max ⁡ { ∑ k = 1 n ( λ ¯ k + d k R + d k I + ε k + γ k + 1 2 ⁢ σ k 2 ⁢ N k 2 ) , 2 ⁢ ∑ k = 1 n ( β k ⁢ k ⁢ N k + β k ⁢ k + γ k ) } .$

Therefore if $t 1 ≤ T$ , there exists

$∫ 0 τ m ∧ t 1 𝑑 V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) ) ≤ ∫ 0 τ m ∧ t 1 [ C ⁢ ( 1 + V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) ) ) ] ⁢ 𝑑 t$

$+ ∫ 0 τ m ∧ t 1 [ ∑ k = 1 n ( σ k ⁢ ( N k - R k ⁢ ( t ) - I k ⁢ ( t ) ) ⁢ ( I k ⁢ ( t ) - 1 ) ) ] ⁢ 𝑑 B k ⁢ ( t ) .$

This implies that

$𝔼 ⁢ [ V ⁢ ( I 1 ⁢ ( τ m ∧ t 1 ) , R 1 ⁢ ( τ m ∧ t 1 ) , … , I n ⁢ ( τ m ∧ t 1 ) , R n ⁢ ( τ m ∧ t 1 ) ) ]$

$≤ V ⁢ ( I 1 ⁢ ( 0 ) , R 1 ⁢ ( 0 ) , … , I n ⁢ ( 0 ) , R n ⁢ ( 0 ) ) + 𝔼 ⁢ [ ∫ 0 τ m ∧ t 1 [ C ⁢ ( 1 + V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) ) ) ] ⁢ 𝑑 t ]$

$≤ V ⁢ ( I 1 ⁢ ( 0 ) , R 1 ⁢ ( 0 ) , … , I n ⁢ ( 0 ) , R n ⁢ ( 0 ) ) + C ⁢ T + C ⁢ 𝔼 ⁢ [ ∫ 0 τ m ∧ t 1 V ⁢ ( I 1 ⁢ ( t ) , R 1 ⁢ ( t ) , … , I n ⁢ ( t ) , R n ⁢ ( t ) ) ⁢ 𝑑 t ]$

$≤ C ⁢ T + C ⁢ ∫ 0 t 1 𝔼 ⁢ [ V ⁢ ( I 1 ⁢ ( τ m ∧ t 1 ) , R 1 ⁢ ( τ m ∧ t 1 ) , … , I n ⁢ ( τ m ∧ t 1 ) , R n ⁢ ( τ m ∧ t 1 ) ) ] ⁢ 𝑑 t$

(A.2)

$+ V ⁢ ( I 1 ⁢ ( 0 ) , R 1 ⁢ ( 0 ) , … , I n ⁢ ( 0 ) , R n ⁢ ( 0 ) ) .$

By the Gronwall inequality, we obtain

$𝔼 ⁢ [ V ⁢ ( I 1 ⁢ ( τ m ∧ T ) , R 1 ⁢ ( τ m ∧ T ) , … , I n ⁢ ( τ m ∧ T ) , R n ⁢ ( τ m ∧ T ) ) ] ≤ C ¯ ,$

where $C ¯ = ( V ⁢ ( I 1 ⁢ ( 0 ) , R 1 ⁢ ( 0 ) , … , I n ⁢ ( 0 ) , R n ⁢ ( 0 ) ) + C ⁢ T ) ⁢ e C ⁢ T$ . Set $Ω m = τ m ≤ T$ for $m ≥ m 1$ and by (A.1), $P ⁢ ( Ω m ) ≥ ϵ$ . We note that for all $ω ∈ Ω m$ , such that $I k ⁢ ( τ m , ω )$ and $R k ⁢ ( τ m , ω )$ , $k = 1 , 2 , … , n$ , equal either m or $1 m$ , and hence $V ⁢ ( I 1 ⁢ ( τ m , ω ) , R 1 ⁢ ( τ m , ω ) , … , I n ⁢ ( τ m , ω ) , R n ⁢ ( τ m , ω ) )$ is no less than

$m + 1 - log ⁡ ( m ) or 1 m + 1 - log ⁡ ( 1 m ) = 1 m + 1 + log ⁡ ( m ) .$

Consequently, there exists

$V ⁢ ( I 1 ⁢ ( τ m ) , R 1 ⁢ ( τ m ) , … , I n ⁢ ( τ m ) , R n ⁢ ( τ m ) ) ≥ [ m + 1 - log ⁡ ( m ) ] ∧ [ 1 m + 1 + log ⁡ ( m ) ] .$

It then follows from (A.1) and (A) that

$C ¯ ≥ 𝔼 ⁢ [ 𝐈 Ω m ⁢ V ⁢ ( I 1 ⁢ ( τ m ) , R 1 ⁢ ( τ m ) , … , I n ⁢ ( τ m ) , R n ⁢ ( τ m ) ) ]$

$≥ ϵ ⁢ { [ m + 1 - log ⁡ ( m ) ] ∧ [ 1 m + 1 + log ⁡ ( m ) ] } ,$

where $𝐈 Ω m$ is the indicator function of $Ω m$ . There is a contradiction $∞ > C ¯ = ∞$ when $m → ∞$ . So we must therefore have $τ ∞ = ∞$ a.s. ∎

References

[1] F. T. Akyildiz and F. S. Alshammari, Complex mathematical SIR model for spreading of COVID-19 virus with Mittag–Leffler kernel, Adv. Difference Equ. 2021 (2021), Paper No. 319. 10.1186/s13662-021-03470-1Search in Google Scholar PubMed PubMed Central

[2] N. Al-Salti, F. Al-Musalhi, I. Elmojtaba and V. Gandhi, SIR model with time-varying contact rate, Int. J. Biomath. 14 (2021), no. 4, Paper No. 2150017. 10.1142/S1793524521500170Search in Google Scholar

[3] L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley & Sons, New York, 1972. Search in Google Scholar

[4] J. Bao and C. Yuan, Convergence rate of EM scheme for SDDEs, Proc. Amer. Math. Soc. 141 (2013), no. 9, 3231–3243. 10.1090/S0002-9939-2013-11886-1Search in Google Scholar

[5] W.-J. Beyn, E. Isaak and R. Kruse, Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, J. Sci. Comput. 70 (2017), no. 3, 1042–1077. 10.1007/s10915-016-0290-xSearch in Google Scholar

[6] L. Chen, S. Gan and X. Wang, First order strong convergence of an explicit scheme for the stochastic SIS epidemic model, J. Comput. Appl. Math. 392 (2021), Paper No. 113482. 10.1016/j.cam.2021.113482Search in Google Scholar

[7] L. Chen and F. Wei, Persistence and distribution of a stochastic susceptible-infected-removed epidemic model with varying population size, Phys. A 483 (2017), 386–397. 10.1016/j.physa.2017.04.114Search in Google Scholar

[8] X. Feng, Z. Teng and F. Zhang, Global dynamics of a general class of multi-group epidemic models with latency and relapse, Math. Biosci. Eng. 12 (2015), no. 1, 99–115. 10.3934/mbe.2015.12.99Search in Google Scholar PubMed

[9] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1976. 10.1016/B978-0-12-268202-5.50014-2Search in Google Scholar

[10] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43 (2001), no. 3, 525–546. 10.1137/S0036144500378302Search in Google Scholar

[11] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2130, 1563–1576. 10.1098/rspa.2010.0348Search in Google Scholar

[12] C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Phys. A. 390 (2011), no. 10, 1747–1762. 10.1016/j.physa.2010.12.042Search in Google Scholar

[13] C. Kahl, M. Günther and T. Rossberg, Structure preserving stochastic integration schemes in interest rate derivative modeling, Appl. Numer. Math. 58 (2008), no. 3, 284–295. 10.1016/j.apnum.2006.11.013Search in Google Scholar

[14] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Appl. Math. (New York) 23, Springer, Berlin, 1992. 10.1007/978-3-662-12616-5Search in Google Scholar

[15] T. Kuniya, J. Wang and H. Inaba, A multi-group SIR epidemic model with age structure, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), 3515–3550. 10.3934/dcdsb.2016109Search in Google Scholar

[16] W. Li, J. Ji, L. Huang and Z. Guo, Global dynamics of a controlled discontinuous diffusive SIR epidemic system, Appl. Math. Lett. 121 (2021), Paper No. 107420. 10.1016/j.aml.2021.107420Search in Google Scholar

[17] X. Li, X. Mao and H. Yang, Strong convergence and asymptotic stability of explicit numerical schemes for nonlinear stochastic differential equations, Math. Comp. 90 (2021), no. 332, 2827–2872. 10.1090/mcom/3661Search in Google Scholar

[18] X. Li, X. Mao and G. Yin, Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment and stability, IMA J. Numer. Anal. 39 (2019), no. 2, 847–892. 10.1093/imanum/dry015Search in Google Scholar

[19] X. Li and H. Yang, Explicit numerical approximation for logistic models with regime switching in finite and infinite horizons, preprint (2021), https://arxiv.org/abs/2106.03540. Search in Google Scholar

[20] X. Li and G. Yin, Explicit Milstein schemes with truncation for nonlinear stochastic differential equations: convergence and its rate, J. Comput. Appl. Math. 374 (2020), Paper No. 112771. 10.1016/j.cam.2020.112771Search in Google Scholar

[21] Q. Liu and D. Jiang, Dynamical behavior of a stochastic multigroup SIR epidemic model, Phys. A 526 (2019), Paper No. 120975. 10.1016/j.physa.2019.04.211Search in Google Scholar

[22] Y. Luo, S. Tang, Z. Teng and L. Zhang, Global dynamics in a reaction-diffusion multi-group SIR epidemic model with nonlinear incidence, Nonlinear Anal. Real World Appl. 50 (2019), 365–385. 10.1016/j.nonrwa.2019.05.008Search in Google Scholar

[23] Y. Luo, L. Zhang, Z. Teng and T. Zheng, Analysis of a general multi-group reaction-diffusion epidemic model with nonlinear incidence and temporary acquired immunity, Math. Comput. Simulation 182 (2021), 428–455. 10.1016/j.matcom.2020.11.002Search in Google Scholar

[24] P. Magal, O. Seydi and G. Webb, Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission, Math. Biosci. 301 (2018), 59–67. 10.1016/j.mbs.2018.03.020Search in Google Scholar PubMed

[25] X. Mao, Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 296 (2016), 362–375. 10.1016/j.cam.2015.09.035Search in Google Scholar

[26] X. Mao and L. Szpruch, Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math. 238 (2013), 14–28. 10.1016/j.cam.2012.08.015Search in Google Scholar

[27] S. Momani, R. Kumar, H. Srivastava, S. Kumar and S. Hadid, A chaos study of fractional SIR epidemic model of childhood diseases, Results. Phys. 27 (2021), Paper No. 104422. 10.1016/j.rinp.2021.104422Search in Google Scholar

[28] X. Mu and Q. Zhang, Near-optimal control for a stochastic multi-strain epidemic model with age structure and Markovian switching, Internat. J. Control 95 (2022), no. 5, 1191–1205. 10.1080/00207179.2020.1843074Search in Google Scholar

[29] A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs defined in a domain, Numer. Math. 128 (2014), no. 1, 103–136. 10.1007/s00211-014-0606-4Search in Google Scholar

[30] A. Suryanto and I. Darti, On the nonstandard numerical discretization of SIR epidemic model with a saturated incidence rate and vaccination, AIMS Math. 6 (2021), no. 1, 141–155. 10.3934/math.2021010Search in Google Scholar

[31] X. Wang, J. Wu and B. Dong, Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition, BIT 60 (2020), no. 3, 759–790. 10.1007/s10543-019-00793-0Search in Google Scholar

[32] S.-L. Wu, L. Chen and C.-H. Hsu, Traveling wave solutions for a diffusive age-structured SIR epidemic model, Commun. Nonlinear Sci. Numer. Simul. 98 (2021), Paper No. 105769. 10.1016/j.cnsns.2021.105769Search in Google Scholar

[33] Y. Xie and C. Zhang, Asymptotical boundedness and moment exponential stability for stochastic neutral differential equations with time-variable delay and Markovian switching, Appl. Math. Lett. 70 (2017), 46–51. 10.1016/j.aml.2017.03.003Search in Google Scholar

[34] H. Yang and J. Huang, First order strong convergence of positivity preserving logarithmic Euler–Maruyama method for the stochastic SIS epidemic model, Appl. Math. Lett. 121 (2021), Paper No. 107451. 10.1016/j.aml.2021.107451Search in Google Scholar

[35] H. Yang and X. Li, Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons, J. Differential Equations 265 (2018), no. 7, 2921–2967. 10.1016/j.jde.2018.04.052Search in Google Scholar

[36] H. Yang, F. Wu, P. E. Kloeden and X. Mao, The truncated Euler-Maruyama method for stochastic differential equations with Hölder diffusion coefficients, J. Comput. Appl. Math. 366 (2020), Paper No. 112379. 10.1016/j.cam.2019.112379Search in Google Scholar

[37] J. Yu, D. Jiang and N. Shi, Global stability of two-group SIR model with random perturbation, J. Math. Anal. Appl. 360 (2009), no. 1, 235–244. 10.1016/j.jmaa.2009.06.050Search in Google Scholar

Received: 2022-07-12

Revised: 2022-10-03

Accepted: 2022-10-20

Published Online: 2022-12-07

Published in Print: 2023-07-01

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DOI: doi.org/10.1515/cmam-2022-0143

Keywords for this article

Stochastic Multi-Group SIR Epidemic Model; Positivity-Preserving; Logarithmic Truncated Euler–Maruyama Method; Strong Convergence