MINIMUM WAVE SPEED OF A REACTION-DIFFUSION DENGUE MODEL WITH ASYMPTOMATIC CARRIER TRANSMISSION

doi:10.1007/s10473-025-0515-x

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (5): 2088-2119.doi: 10.1007/s10473-025-0515-x

MINIMUM WAVE SPEED OF A REACTION-DIFFUSION DENGUE MODEL WITH ASYMPTOMATIC CARRIER TRANSMISSION

Qin XING^1,2,3, Rui XU^1,2,*

1. Complex Systems Research Center, Shanxi University, Taiyuan 030006, China;
2. Complex Systems and Data Science Key Laboratory of Ministry of Education, Shanxi University, Taiyuan 030006, China;
3. School of Mathematics and Statistics, Shanxi University, Taiyuan 030006, China

收稿日期:2024-03-04 修回日期:2024-06-23 出版日期:2025-09-25 发布日期:2025-10-14

MINIMUM WAVE SPEED OF A REACTION-DIFFUSION DENGUE MODEL WITH ASYMPTOMATIC CARRIER TRANSMISSION

Qin XING^1,2,3, Rui XU^1,2,*

1. Complex Systems Research Center, Shanxi University, Taiyuan 030006, China;
2. Complex Systems and Data Science Key Laboratory of Ministry of Education, Shanxi University, Taiyuan 030006, China;
3. School of Mathematics and Statistics, Shanxi University, Taiyuan 030006, China

Received:2024-03-04 Revised:2024-06-23 Online:2025-09-25 Published:2025-10-14
Contact: ^*Rui Xu, E-mail: rxu88@163.com
About author:Qin Xing, E-mail: xingqin5103@163.com
Supported by:
Xu's research was supported by the National Natural Science Foundation of China (12271317, 11871316).

摘要/Abstract

摘要： Dengue is a mosquito-borne disease that is rampant worldwide, with up to 70\% of cases reported to be asymptomatic during epidemics. In this paper, a reaction-diffusion dengue model with asymptomatic carrier transmission is investigated. We aim to study the existence, nonexistence and minimum wave speed of traveling wave solutions to the model. The results show that the existence and nonexistence of traveling wave solutions are fully determined by the threshold values, which are, the basic reproduction number $R_0$ and critical wave speed $c^*>0$. Specifically, when $R_0>1$ and the wave speed $c\ge c^*$, the existence of the traveling wave solution is obtained by using Schauder's fixed point theorem and Lyapunov functional. It is proven that the model has no nontrivial traveling wave solutions for $R_0\le1$ or $R_0>1$ and $0<c<c^*$ by employing comparison principle and limit theory. As a consequence, we conclude that the critical wave speed $c^*$ is the minimum wave speed of the model. Finally, numerical simulations are carried out to illustrate the effects of several important parameters on the minimum wave speed.

关键词: dengue, asymptomatic carriers, reaction-diffusion, traveling wave solutions, minimum wave speed

Abstract: Dengue is a mosquito-borne disease that is rampant worldwide, with up to 70\% of cases reported to be asymptomatic during epidemics. In this paper, a reaction-diffusion dengue model with asymptomatic carrier transmission is investigated. We aim to study the existence, nonexistence and minimum wave speed of traveling wave solutions to the model. The results show that the existence and nonexistence of traveling wave solutions are fully determined by the threshold values, which are, the basic reproduction number $R_0$ and critical wave speed $c^*>0$. Specifically, when $R_0>1$ and the wave speed $c\ge c^*$, the existence of the traveling wave solution is obtained by using Schauder's fixed point theorem and Lyapunov functional. It is proven that the model has no nontrivial traveling wave solutions for $R_0\le1$ or $R_0>1$ and $0<c<c^*$ by employing comparison principle and limit theory. As a consequence, we conclude that the critical wave speed $c^*$ is the minimum wave speed of the model. Finally, numerical simulations are carried out to illustrate the effects of several important parameters on the minimum wave speed.

Key words: dengue, asymptomatic carriers, reaction-diffusion, traveling wave solutions, minimum wave speed

中图分类号:

35C07

Qin XING, Rui XU. MINIMUM WAVE SPEED OF A REACTION-DIFFUSION DENGUE MODEL WITH ASYMPTOMATIC CARRIER TRANSMISSION[J]. 数学物理学报(英文版), 2025, 45(5): 2088-2119.

Qin XING, Rui XU. MINIMUM WAVE SPEED OF A REACTION-DIFFUSION DENGUE MODEL WITH ASYMPTOMATIC CARRIER TRANSMISSION[J]. Acta mathematica scientia,Series B, 2025, 45(5): 2088-2119.

参考文献

[1] Simmons C P, Farrar J J, Nguyen V C, Wills B. Dengue. N Engl J Med, 2012, 366(15): 1423-1432
[2] Kyle J L, Harris E. Global spread and persistence of dengue. Annu Rev Microbiol, 2008, 62(1): 71-92
[3] World Health Organization.Dengue and severe dengue. available on: https://www.who.int/zh/news-room/fact-sheets/detail/dengue-and-severe-dengue
[4] Wilder-Smith A. Dengue vaccine development: status and future. Bundesgesundheitsbl, 2020, 63: 40-44
[5] Thomas S J. Is new dengue vaccine efficacy data a relief or cause for concern?. NPJ Vaccines, 2023, 8(1): Art 55
[6] Esteva L, Vargas C. Analysis of a dengue disease transmission model. Math Biosci, 1998, 150(2): 131-151
[7] Burke D S, Nisalak A, Johnson D E. A prospective study of dengue infections in Bangkok. Am J Trop Med Hyg, 1988, 38: 172-180
[8] Endy T P, Chunsuttiwat S, Nisalak A, et al. Epidemiology of inapparent and symptomatic acute dengue virus infection: a prospective study of primary school children in Kamphaeng Phet, Thailand. Am J Epidemiol, 2002, 156(1): 40-51
[9] Baaten G G G, Sonder G J B, Zaaijer H L, et al. Travel-related dengue virus infections, the Netherlands, 2006-2007. Emerging Infect Dis, 2011, 17(5): 821-827
[10] Duong V, Lambrechts L, Paul R E, et al. Asymptomatic humans transmit dengue virus to mosquitoes. Proc Natl Acad Sci, 2015, 112(47): 14688-14693
[11] Ten Bosch Q A, Clapham H E, Lambrechts L, et al. Contributions from the silent majority dominate dengue virus transmission. PLoS Pathog, 2018, 14(4): e1006965
[12] Grunnill M. An exploration of the role of asymptomatic infections in the epidemiology of dengue viruses through susceptible, asymptomatic, infected and recovered (SAIR) models. J Theor Biol, 2018, 439: 195-204
[13] Jan R, Khan M A, Gómez-Aguilar J F. Asymptomatic carriers in transmission dynamics of dengue with control interventions. Optim Contr Appl Met, 2020, 41: 430-447
[14] Rigau-Perez J G, Gubler D J, Vorndam, Clark G G. Dengue: a literature review and case study of travelers from the United States, 1986-1994. J Travel Med, 1997, 4(2): 65-71
[15] Adams B, Kapan D D. Man bites mosquito: understanding the contribution of human movement to vector-borne disease dynamics. PLoS One, 2009, 4(8): e6763
[16] Guzman M G, Halstead S B, Artsob H, et al. Dengue: a continuing global threat. Nat Rev Microbiol, 2010, 8: 7-16
[17] Getis A, Morrison A C, Gray K, et al. Characteristics of the spatial pattern of the dengue vector, Aedes aegypti, in Iquitos, Peru. Am J Trop Med Hyg, 2003, 69: 494-505
[18] Thai K T D, Anders K L. The role of climate variability and change in the transmission dynamics and geographic distribution of dengue. Exp Biol Med, 2011, 263: 944-54
[19] Liang X, Zhao X Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with application. Commun Pur Appl Math, 2007, 60: 1-40
[20] Li T, Li Y, Hethcote H W. Periodic traveling waves in SIRS model. Math Comp Model Dyn, 2009, 49: 393-401
[21] Wu C C. Existence of traveling waves with the critical speed for a discrete diffusive epidemic model. J Differ Equ, 2017, 262: 272-282
[22] Wang Z C, Wu J, Liu R, Traveling waves of the spread of avian influenza. Proc Am Math Soc, 2012, 140: 3931-3946
[23] Li Y, Li W T, Lin G. Traveling waves of a delayed diffusive SIR epidemic model. Commun Pure Appl Anal, 2015, 14: 1001-1022
[24] Wang K, Zhao H Y, Wang H, et al. Traveling wave of a reaction-diffusion vector-borne disease model with nonlocal effects and distributed delay. J Dyn Differ Equ, 2021, 35: 3149-3185
[25] Zhao L, Wang Z C, Ruan S.Traveling wave solutions in a two-group SIR epidemic model with constant recruitment. J Math Biol, 2018, 77: 1871-1915
[26] Zhao L, Wang Z C, Ruan S. Traveling wave solutions in a two-group epidemic model with latent period. Nonlinearity, 2017, 30: 1287-1325
[27] Denu D, Ngoma S, Salako R B. Existence of traveling wave solutions of a deterministic vector-host epidemic model with direct transmission. J Math Anal Appl, 2020, 487: Art 123995
[28] Wang K, Zhao H, Wang H. Traveling waves for a diffusive mosquito-borne epidemic model with general incidence. Z Angew Math Phys, 2022, 73: Art 31
[29] Zhao L. Spreading speed and traveling wave solutions of a reaction-diffusion Zika model with constant recruitment. Nonlinear Anal: RWA, 2023, 74: Art 103942
[30] Gazori F, Hesaaraki M. Minimum wave speed for dengue prevalence in the symptomatic and asymptomatic infected individuals. Comput Appl Math, 2023, 42: 1-23
[31] Yang Z, Zhang B G. Global stability of traveling wavefronts for nonlocal reaction-diffusion equations with time delay. Acta Math Sci, 2018, 38: 289-302
[32] Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci, 2002, 180: 29-48
[33] Ma S. Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem. J Differ Equ, 2001, 171: 294-314
[34] Gilbarg G, Trudinger N.Elliptic Partial Differential Equations of Second Order. Berlin: Springer, 2001
[35] Friedman A.Partial Differential Equations of Parabolic Type. Courier Dover Publications, 2008
[36] Lam K Y, Lou Y. Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications. J Dyn Differ Equ, 2016, 28: 29-48

MINIMUM WAVE SPEED OF A REACTION-DIFFUSION DENGUE MODEL WITH ASYMPTOMATIC CARRIER TRANSMISSION

MINIMUM WAVE SPEED OF A REACTION-DIFFUSION DENGUE MODEL WITH ASYMPTOMATIC CARRIER TRANSMISSION

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 11

Metrics

本文评价

推荐阅读 0

[1]	Huijuan ZHU, Xiaojun LI, Yanjiao LI. EXPONENTIAL STABILITY OF ATTRACTOR FOR RANDOM REACTION-DIFFUSION EQUATION DRIVEN BY NONLINEAR COLORED NOISE ON ℝ^N[J]. 数学物理学报(英文版), 2025, 45(4): 1567-1596.
[2]	Hao HAN, Chengjian ZHANG. ASYMPTOTICAL STABILITY OF NEUTRAL REACTION-DIFFUSION EQUATIONS WITH PCAS AND THEIR FINITE ELEMENT METHODS^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1865-1880.
[3]	胡淑兰, 李瑞囡, 王新宇. CENTRAL LIMIT THEOREM AND MODERATE DEVIATIONS FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2020, 40(5): 1477-1494.
[4]	Lamia DJEBARA, Salem ABDELMALEK, Samir BENDOUKHA. GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR SOME COUPLED SYSTEMS VIA A LYAPUNOV FUNCTIONAL[J]. 数学物理学报(英文版), 2019, 39(6): 1538-1550.
[5]	杨兆星, 张国宝. GLOBAL STABILITY OF TRAVELING WAVEFRONTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY[J]. 数学物理学报(英文版), 2018, 38(1): 289-302.
[6]	钟延生, 孙春友. UNIFORM QUASI-DIFFERENTIABILITY OF SEMIGROUP TO NONLINEAR BEACTION-DIFFUSION EQUATIONS WITH SUPERCRITICAL EXPONENT[J]. 数学物理学报(英文版), 2017, 37(2): 301-315.
[7]	蒋咪娜, 向建林. ASYMPTOTIC STABILITY OF TRAVELING WAVES FOR A DISSIPATIVE NONLINEAR EVOLUTION SYSTEM[J]. 数学物理学报(英文版), 2015, 35(6): 1325-1338.
[8]	Xinfeng Liu, Qing Nie. SPATIALLY-LOCALIZED SCAFFOLD PROTEINS MAY FACILITATE TO TRANSMIT LONG-RANGE SIGNALS[J]. 数学物理学报(英文版), 2009, 29(6): 1657-1669.
[9]	金海, 李正元, 叶其孝. POSITIVENESS THEOREMS OF SOLUTIONS FOR SEVERAL DIFFERENTIAL INEQUALITIES AND THEIR APPLICATIONS[J]. 数学物理学报(英文版), 2002, 22(2): 227-240.
[10]	陈登远, 谷元. COLE-HOPF QUOTIENT AND EXACT SOLUTIONS OF THE GENERALIZED FITZHUGH-NAGUMO EQUATIONS[J]. 数学物理学报(英文版), 1999, 19(1): 7-14.
[11]	江成顺, 李海峰. GLOBAL EXISTENCE OF SOLUTIONS FOR A STRONGLY COUPLED REACTION-DIFFUSION SYSTEM[J]. 数学物理学报(英文版), 1998, 18(1): 1-10.