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.
[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
