Critical traveling waves in a diffusive disease model

Abstract

In this paper, the existence of a non-trivial, positive and bounded critical traveling wave solution of a diffusive disease model, whose reaction system has infinity many equilibria, is obtained for the first time. This gives an affirmative answer to an open problem left in [X. Wang, H. Wang, J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Syst. Ser. A 32 (2012) 3303-3324]. Our result shows that the critical traveling wave in this model is a mixed of front and pulse type.

Critical traveling wave solutions for a vaccination model with general incidence

“…In contrast to the non-critical traveling wave solutions, the existence problem of critical traveling wave solutions is more difficult in various models. In past years, using limiting arguments or Schauder's fixed point theorem, there were some literature devoting to investigate the existence of critical traveling wave solutions of different models, see e.g., [2,3,7,8,10,13,14]. Applying the limiting arguments, it is sufficient to consider the convergence problem for a sequence of traveling wave solutions with wave speed greater than the critical speed.…”

“…However, the convergence problem is non-trivial. Motivated by the literature [13,15], in this work we establish a pair of upper-and lower-solutions of system (3) and apply the Schauder's fixed point theorem to prove the existence of critical traveling wave solutions of (1) with general incidence.…”

Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay

“…In diffusive epidemic models, traveling waves can describe the state that a disease spreads geographically with a constant speed. The existence of traveling waves in these models has become one of the important issues in mathematical epidemiology [1,5,[7][8][9]11,11,12,14,20,[27][28][29][30][31][32][33]35,36,38,38,[40][41][42][43][44][45][46][48][49][50]. For example, Wang et al [30] where S(x, t), I(x, t) and R(x, t) refer to the densities of susceptible, infected and recovered individuals at location x and time t, respectively.…”

“…In this paper, inspired by [11], we will apply the upper-lower solution method together with Schauder's fixed point theorem to investigate the existence of critical traveling waves for (1.7). This upper-lower solution method is concise and explicit, and has been used subsequently to other diffusive models [12,22,48]. We note that the research on the existence of critical traveling wave solutions to nonlocal dispersal epidemic models with spatio-temporal delay have not been seen in the literature.…”

“…Critical traveling waves. Motivated by [11,12,22,48], in this section we will prove the existence result for R 0 > 1 and c = c * in Theorem 1.2. For our purpose, we choose L 3 > 1 to be suitable large such that the equation −L 3 ze λ * z = (β−γ−δ)S1 γ+δ admits two negative roots z 4 and z * satisfying…”