朱春娟
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一類具有飽和傳染率、免疫接種和垂直傳染的SIR傳染病模型的全局穩定性分析
朱春娟
(韶關學院數學與統計學院,廣東,韶關 512005)
針對一類具有飽和傳染率、免疫接種和垂直傳染的SIR傳染病模型,確定了疾病的基本再生數。得出結論:當疾病的基本再生數小于1時,無病平衡點是全局漸近穩定的,當疾病基本再生數大于1時,地方病平衡點是全局漸近穩定的。
SIR傳染病模型;飽和傳染率;免疫接種;垂直傳染;穩定性
傳染病在現實生活中廣泛存在,利用動力學方法來研究傳染病是非常重要的方法之一[1-3]。在文獻[4-9]中研究關于垂直傳染的傳染病模型,這里認為若新生兒的母親為染病者,而新生兒卻不能接種,但事實上,有些是可以接種的,譬如乙肝等。
基于上述考慮,本文研究了更符合實際生物情況的具有飽和傳染率、免疫接種和垂直傳染的SIR傳染病模型。
為了更符合實際,我們建立的模型中有帶有免疫接種的常數輸入者,且考慮垂直傳染。我們主要討論以下模型:
結合(2),系統(1)等價于系統
利用再生矩陣的方法,可以得到系統(3)的基本再生數為
把(4)代入系統(3)的第一個方程得
定義
則
由于
則
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Global Stability of a SIR Epidemic Model with Saturating Infect Rate,Vaccination and Vertical Transmission
ZHU Chun-juan
(College of Mathematics and Statistics, Shaoguan University, Shaoguan, Guangdong 512005,China)
In this paper, the SIR epidemic model with saturating infect rate, vaccination and vertical transmission is studied; the basic reproduction numberfor the existence of infectivedisease was established. If, the disease-free equilibrium is global asymptotically stable. If, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region .We discuss the biological significance at the end.
SIR epidemic model; saturating infect rate; vaccination; vertical transmission; equilibrium; global stability
1674-8085(2015)04-0013-04
O175.13
A
10.3969/j.issn.1674-8085.2015.04.003
2015-04-08;修改日期:2015-05-24
韶關市科技計劃項目(2014CX/K231)
朱春娟(1982-),女,江蘇南通人,講師,碩士,主要從事生物數學方向研究(zcjyangyang.sina.com).