于智慧，王錚

帶有短暫免疫的分?jǐn)?shù)階SIRS模型的局部穩(wěn)定性

于智慧1，王錚2

（營(yíng)口理工學(xué)院1. 基礎(chǔ)教研部，2.化學(xué)與環(huán)境工程學(xué)院，遼寧 營(yíng)口 115014）

研究了帶有短暫免疫的分?jǐn)?shù)階SIRS模型，在模型中，當(dāng)個(gè)體一旦被感染，在一段時(shí)間后會(huì)轉(zhuǎn)向?yàn)橐赘袀€(gè)體．討論了該模型解的正則有界性及解的唯一性，給出了在零染病平衡點(diǎn)及有染病平衡點(diǎn)處系統(tǒng)局部漸進(jìn)穩(wěn)定的充分條件．

SIRS模型；隨機(jī)平均法；平衡點(diǎn)；穩(wěn)定

近些年來(lái)，SIR，SEI，SEIR和SIRS流行病模型的動(dòng)力學(xué)已經(jīng)獲得了廣泛的關(guān)注[1-10]．文獻(xiàn)[11]提供了２個(gè)時(shí)延的不同的SEIRS模型，并且研究了平衡點(diǎn)的穩(wěn)定性．文獻(xiàn)[12]提出了喪失疫苗免疫的時(shí)延SIRS流行病模型，給出了整數(shù)階的帶有短暫免疫的SIRS模型的全局穩(wěn)定性分析．文獻(xiàn)[13]給出了整數(shù)階的帶有短暫免疫的SIRS模型的全局穩(wěn)定性分析．文獻(xiàn)[14]得到了疾病的非線性發(fā)病率和短暫免疫的模型，對(duì)該模型進(jìn)行了定量的分析.

本文將文獻(xiàn)[13]的模型進(jìn)行推廣，給出了帶有疾病的短暫免疫和線性發(fā)病率的分?jǐn)?shù)階模型．相比于整數(shù)階模型，分?jǐn)?shù)階模型更具有動(dòng)態(tài)記憶和遺傳性，應(yīng)用得更廣泛．

1 模型公式和正解

考慮帶有短暫免疫的分?jǐn)?shù)階SIRS時(shí)延模型

初始條件為

2　零染病平衡點(diǎn)及其穩(wěn)定性

系統(tǒng)（3）的平衡點(diǎn)滿足系統(tǒng)

定義再生感染數(shù)為

3 有染病平衡點(diǎn)及其穩(wěn)定性

此時(shí)沒(méi)有零染病平衡點(diǎn)．

在條件（8）下，對(duì)應(yīng)特征方程（5）有方程

對(duì)式（11）平方再加和，可以得到

令

4 結(jié)語(yǔ)

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[17] Hale J K．Theory of functional differential equations[M]．New York：Springer，1978：610-612．

Local stability of a fractional-order SIRS model with temporary immunity

YU Zhihui1，WANG Zheng2

（1. Basic Courses Teaching and Research Department，2. School of Chemical and Environmental Engineering，Yingkou Institute of Technology，Yingkou 115014，China）

The fractional-order SIRS model with temporary immunity is studied．In the model，once an individual is infected，the individual will turn to susceptible individuals after a period of time．The regular boundedness and uniqueness of the solution of the model are discussed．Sufficient conditions for the local asymptotic stability of the system at the zero infection equilibrium point and the infection equilibrium point are given．

SIRS model；stochastic averaging method；equilibrium；stability

1007-9831（2022）11-0022-04

O175

A

10.3969/j.issn.1007-9831.2022.11.004

2022-04-01

校級(jí)科研項(xiàng)目（QNL202104）

于智慧（1991-），女，遼寧朝陽(yáng)人，助教，碩士，從事應(yīng)用數(shù)學(xué)研究．E-mail：yuzhihuijiayou@163.com