范佳欣,孫福芹
具有無癥狀感染者和病毒變異的傳染病模型
范佳欣,孫福芹
(天津職業(yè)技術(shù)師范大學 理學院,天津 300222)
研究了一類具有無癥狀感染者和病毒變異的傳染病模型.計算了模型的基本再生數(shù),求出了模型的無病平衡點和地方病平衡點.根據(jù)Jacobi矩陣特征根的符號,利用Lyapunov函數(shù)法,分析了基本再生數(shù)不同取值時無病平衡點和地方病平衡點的穩(wěn)定性情況.通過數(shù)值模擬對結(jié)論進行了驗證.
無癥狀感染者;病毒變異;傳染病模型;穩(wěn)定性
傳染病模型是數(shù)學在生物學中應用的一個途徑,為人類了解傳染病特性提供了一定的幫助[1-8].近幾年爆發(fā)的新冠肺炎疫情再次引起了人們對于傳染病的關(guān)注,新冠病毒自身存在的復雜的變異情況是新冠疫情蔓延廣泛而持久的一個重要原因.為了更好地了解存在病毒變異的傳染病的傳播情況,需要對具有病毒變異情況的傳染病模型進行研究.文獻[9]對具有病毒變異的SEIR傳染病模型進行了分析.而在新冠肺炎疫情中,存在不表現(xiàn)出臨床癥狀,卻能夠傳播感染的無癥狀感染者,這與潛伏期患者不同,這樣的特性也將對疫情的傳播產(chǎn)生一定的影響.本文在具有病毒變異的傳染病模型基礎(chǔ)上,考慮無癥狀感染者對于傳染病傳播的影響,建立一類具有無癥狀感染者和病毒變異的傳染病模型,并分析其平衡點的穩(wěn)定性.
基于病毒發(fā)生變異情況的傳染病模型,考慮無癥狀感染者對病毒傳播的影響,建立一類具有無癥狀感染者和病毒變異的傳染病模型,即
式中:分別為時刻易感人群、無癥狀感染人群、病毒變異前的感染人群、病毒變異后的感染人群和移除人群在總?cè)巳褐械恼急?,無癥狀感染者是指表示不表現(xiàn)出臨床癥狀,但能夠傳播感染的傳染病患者;為出生率和自然死亡率;,,分別為無癥狀感染者、病毒變異前的感染者和病毒變異后的感染者的有效接觸率;,,分別為無癥狀感染者、病毒變異前的感染者和病毒變異后的感染者的移除率;為無癥狀感染者的癥狀發(fā)作率;為病毒變異前的感染者轉(zhuǎn)換為病毒變異后的感染者的比率.模型(1)的流程見圖1,初始條件為,并屬于集合.
將式(5)~(8)代入式(2)中,可得
圖2 無病平衡點和地方病平衡點的穩(wěn)定性
由圖2可以看出,數(shù)值模擬結(jié)果與定理3~4結(jié)論相符.
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Epidemic model with asymptomatic infection and virus mutation
FAN Jiaxin,SUN Fuqin
(School of Sciences,Tianjin University of Technology and Education,Tianjin 300222,China)
A class of epidemic model with asymptomatic infection and virus mutation was studied.The basic reproduction number of the model were calculated,the disease-free equilibrium point and endemic equilibrium point of the model were obtained.Then,the sign of the characteristic root of Jacobi matrix and the method of Lyapunov functions were used to analys the stability of the disease-free equilibrium point and endemic equilibrium point with different values of basic reproduction number.Finally,the obtained results were verified by numerical simulation.
asymptomatic infection;virus mutation;epidemic model;stability
1007-9831(2023)09-0009-07
O175.13
A
10.3969/j.issn.1007-9831.2023.09.003
2023-01-03
范佳欣(1998-),女,江西新余人,在讀碩士研究生,從事生物數(shù)學研究.E-mail:fanjiaxin.1118@qq.com
孫福芹(1970-),男,山東單縣人,教授,博士,從事偏微分方程及生物數(shù)學研究.E-mail:sfqwell@163.com