The Numerical Approximation to a Stochastic Age-Structured HIV/AIDS Model with Nonlinear Incidence Rates

Jie Ren; Huaimin Yuan; Qimin Zhang

doi:10.1515/cmam-2021-0154

Published by De Gruyter March 9, 2022

The Numerical Approximation to a Stochastic Age-Structured HIV/AIDS Model with Nonlinear Incidence Rates

Jie Ren , Huaimin Yuan and Qimin Zhang

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2021-0154

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Abstract

In this paper, a stochastic age-structured HIV/AIDS model with nonlinear incidence rates is proposed. It is of great importance to develop efficient numerical approximation methods to solve this HIV/AIDS model since most stochastic partial differential equations (SPDEs) cannot be solved analytically. From the perspective of biological significance, the exact solution of the HIV/AIDS model must be nonnegative and bounded. Then a modified explicit Euler–Maruyama (EM) scheme is constructed based on a projection operator. The EM scheme could preserves the nonnegativity of the numerical solutions and also make the numerical solutions not outside the domain of the exact solutions. The convergence results between the numerical solutions and the exact solutions are analyzed, and some numerical examples are given to verify our theoretical results.

Keywords: Stochastic Age-Structured HIV/AIDS Model; Explicit Euler–Maruyama Scheme; Projection Operator; Convergence

MSC 2010: 60H35; 65C20; 92B05

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12161068

Funding source: Natural Science Foundation of Ningxia Province

Award Identifier / Grant number: 2020AAC03065

Award Identifier / Grant number: 2021AAC03022

Funding statement: The research was supported in part by the National Natural Science Foundation of China (No. 12161068) and by the Natural Science Foundation of Ningxia Province (CN) (grant numbers 2020AAC03065 and 2021AAC03022).

A Proof of Theorem 3.2

Proof

Summing the equations of system (2.1), we have

d t ⁢ ( S + I + A ) = { - ∂ ⁡ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] ∂ ⁡ a - μ ⁢ ( t ) ⁢ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] - μ 2 ⁢ ( t , a ) ⁢ A ⁢ ( t , a ) } ⁢ d ⁢ t - σ ⁢ ( t ) ⁢ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] ⁢ d ⁢ B t ≤ { - ∂ ⁡ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] ∂ ⁡ a - μ ⁢ ( t ) ⁢ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] } ⁢ d ⁢ t - σ ⁢ ( t ) ⁢ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] ⁢ d ⁢ B t .

Let U ⁢ ( t , a ) := S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) . According to the comparison theorem, the boundedness problem of system (2.1) is transformed into the corresponding problem of the following linear age-structured model with multiplicative noise:

(A.1) { d t ⁢ U = [ - ∂ ⁡ U ⁢ ( t , a ) ∂ ⁡ a - μ ⁢ ( t ) ⁢ U ⁢ ( t , a ) ] ⁢ d ⁢ t - σ ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d ⁢ B t in ⁢ Q , U ⁢ ( t , 0 ) = ∫ 0 a ¯ β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d a , t ∈ [ 0 , T ] , U ⁢ ( 0 , a ) = U 0 ⁢ ( a ) , a ∈ [ 0 , a ¯ ] ,

where d t ⁢ U := ∂ ⁡ U ⁢ ( t , a ) ∂ ⁡ t ⁢ d ⁢ t . By integration along the characteristic lines and using the variation of constants formula, the explicit solution of equation (A.1) is as follows (see [36]):

(A.2) U ⁢ ( t , a ) = { U 0 ⁢ ( a - t ) ⁢ e ∫ 0 t - μ ⁢ ( s ) ⁢ d ⁢ s - ∫ 0 t σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d ⁢ B s , 0 ≤ t ≤ a , U ⁢ ( t - a , 0 ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s - ∫ t - a t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s , 0 ≤ a < t .

Since U ⁢ ( t , 0 ) = ∫ 0 a ¯ β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d a , from (A.2), we get

(A.3) U ⁢ ( t , 0 ) = ∫ 0 a ¯ β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d ⁢ a = ∫ 0 t β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d ⁢ a + ∫ t t ¯ β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d ⁢ a = J ⁢ ( t ) + ∫ 0 t β ⁢ ( t ) ⁢ U ⁢ ( t - a , 0 ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s ⁢ d a ,

where

J ⁢ ( t ) = ∫ t a ¯ β ⁢ ( a ) ⁢ U 0 ⁢ ( a - t ) ⁢ e ∫ 0 t - μ ⁢ ( s ) ⁢ d ⁢ s ⁢ d ⁢ a - ∫ 0 t β ⁢ ( t ) ⁢ ∫ t - a t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d ⁢ B s ⁢ d ⁢ a - ∫ t a ¯ β ⁢ ( t ) ⁢ ∫ 0 t σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s ⁢ d a , 0 ≤ t ≤ a ¯ ,

and

(A.4) K ⁢ ( t , a ) = β ⁢ ( t ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s .

Since

E ⁢ ( ∫ 0 t σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s ) = E ⁢ ( ∫ t - a t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s ) = 0 ,

from (A.2), we know that the first moment of the solution to equation (A.1) is

(A.5) E ⁢ [ U ⁢ ( t , a ) ] = { U 0 ⁢ ( a - t ) ⁢ e ∫ 0 t - μ ⁢ ( s + t - a ) ⁢ d ⁢ s , 0 ≤ t ≤ a , E ⁢ [ U ⁢ ( t - a , 0 ) ] ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s , 0 ≤ a < t .

By (A.3), we have

E ⁢ [ U ⁢ ( t , 0 ) ] = E ⁢ [ J ⁢ ( t ) ] + ∫ 0 t K ⁢ ( t , a ) ⁢ E ⁢ U ⁢ ( t - a , 0 ) ⁢ d a = E ⁢ [ J ⁢ ( t ) ] + ∫ 0 t K ⁢ ( t , t - s ) ⁢ E ⁢ U ⁢ ( s , 0 ) ⁢ d s ,

where

(A.6) E ⁢ [ J ⁢ ( t ) ] = ∫ t a ¯ β ⁢ ( t ) ⁢ U 0 ⁢ ( a - t ) ⁢ e ∫ 0 t - μ ⁢ ( s ) ⁢ d ⁢ s ⁢ d a .

From equations (A.4), (A.6) and assumptions 1, 2, it follows that

0 ≤ K ⁢ ( t , a ) ≤ K ¯ , ( t , a ) ∈ Q ; 0 ≤ E ⁢ [ J ⁢ ( t ) ] ≤ J ¯ , t ∈ ( 0 , T ) ,

Using a method similar to [2, Theorem 2.1.1], we may infer that E ⁢ [ U ⁢ ( t , a ) ] ≤ C 1 , where C 1 is a positive constant.

We now discuss the second moment of U ⁢ ( t , a ) . According (A.2) and (A.5), we get

(A.7) U ⁢ ( t , a ) - E ⁢ [ U ⁢ ( t , a ) ] = { - ∫ 0 t σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s , 0 ≤ t ≤ a , ( U ⁢ ( t - a , 0 ) - E ⁢ [ U ⁢ ( t - a , 0 ) ] ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s - ∫ t - a t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s , 0 ≤ a < t .

When 0 ≤ t ≤ a , the second moment of U ⁢ ( t , a ) is

M ⁢ ( t , a ) ≜ E ⁢ ( U ⁢ ( t , a ) - E ⁢ [ U ⁢ ( t , a ) ] ) 2 = E ⁢ ∫ 0 t ( σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ) 2 ⁢ d s ≤ σ ¯ 2 ⁢ ∫ 0 t E ⁢ | U ⁢ ( s , a ) | 2 ⁢ d s .

Hence

E ⁢ | U ⁢ ( t , a ) | 2 ≤ ( E ⁢ | U ⁢ ( t , a ) | ) 2 + σ ¯ 2 ⁢ ∫ 0 t E ⁢ | U ⁢ ( s , a ) | 2 ⁢ d s .

By Gronwall’s lemma, we have

E ⁢ | U ⁢ ( t , a ) | 2 ≤ ( E ⁢ | U ⁢ ( t , a ) | ) 2 ⁢ e σ ¯ 2 ⁢ T ≤ C 2 ⁢ e σ ¯ 2 ⁢ T .

When 0 ≤ a < t , the second moment of U ⁢ ( t , a ) is

M ⁢ ( t , a ) ≜ E ⁢ ( U ⁢ ( t , a ) - E ⁢ [ U ⁢ ( t , a ) ] ) 2 = M ⁢ ( t - a , 0 ) ⁢ e ∫ 0 a - 2 ⁢ μ ⁢ ( s + t - a ) ⁢ d ⁢ s + ∫ t - a t σ 2 ⁢ ( s + a - t ) ⁢ e ∫ s t - 2 ⁢ μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ | U ⁢ ( s , s + a - t ) | 2 ⁢ d ⁢ s - 2 E [ ( U ( t - a , 0 ) - E [ U ( t - a , 0 ) ] ) e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s ⋅ ∫ t - a t σ ( s + a - t ) e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ U ( s , s + a - t ) d B s ] ,

where M ⁢ ( t , 0 ) ≜ E ⁢ ( U ⁢ ( t , 0 ) - E ⁢ [ U ⁢ ( t , 0 ) ] ) 2 . It is necessary to study the boundedness of M ⁢ ( t , 0 ) . From (A.2), we have

(A.8) U ⁢ ( t , 0 ) - E ⁢ [ U ⁢ ( t , 0 ) ] = G ⁢ ( t ) + ∫ 0 t K ⁢ ( t , a ) ⁢ [ U ⁢ ( t - a , 0 ) - E ⁢ U ⁢ ( t - a , 0 ) ] ⁢ d a ,

where K ⁢ ( t , a ) = β ⁢ ( t ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s , and

G ⁢ ( t ) = - ∫ 0 t β ⁢ ( t ) ⁢ ∫ t - s t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d a ⁢ d B s - ∫ 0 t β ⁢ ( t ) ⁢ ∫ t a ¯ σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d a ⁢ d B s .

Then

M ⁢ ( t , 0 ) ≤ 2 ⁢ E ⁢ { [ G ⁢ ( t ) ] 2 + ( ∫ 0 t K ⁢ ( t , t - s ) ⁢ [ U ⁢ ( s , 0 ) - E ⁢ U ⁢ ( s , 0 ) ] ⁢ d a ) 2 } .

Under assumptions 1, 2 and by Gronwall’s lemma, we may infer that M ⁢ ( t , 0 ) ≤ C 2 for all ( t , a ) ∈ Q , where C 2 is a positive constant. Combining (A.7) and (A.8), we may have E ⁢ ( U ⁢ ( t , a ) - E ⁢ [ U ⁢ ( t , a ) ] ) 2 ≤ C 3 for all ( t , a ) ∈ Q , where C 3 is a positive constant. Due the boundedness of E ⁢ [ U ⁢ ( t , a ) ] , we may infer that

E ⁢ | S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) | 2 = E ⁢ | U ⁢ ( t , a ) | 2 ≤ C 4 for all ⁢ ( t , a ) ∈ Q ,

where C 4 is a positive constant. Due to the nonnegativity of solutions for system (2.1), we have

E ⁢ ( | S ⁢ ( t , a ) | 2 + | I ⁢ ( t , a ) | 2 + | A ⁢ ( t , a ) | 2 ) ≤ C .

The proof is complete. ∎

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Received: 2021-08-18

Revised: 2022-01-20

Accepted: 2022-01-26

Published Online: 2022-03-09

Published in Print: 2022-07-01

The Numerical Approximation to a Stochastic Age-Structured HIV/AIDS Model with Nonlinear Incidence Rates

Abstract

A Proof of Theorem 3.2

Proof

References

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