Traveling waves in the Kermack–McKendrick epidemic model with latent period

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 the real world, many diseases can't be transmitted to others immediately after being infected and have a latent period. He et al [12] studied the discrete time delayed version of (1.1) where τ ≥ 0 is the time delay. They showed that there exists a constant c * > 0 such that (1.3) admits a nonnegative traveling wave solution (S(x + ct), I(x + ct), R(x + ct)) satisfying boundary conditions (1.2) when R 0 = β/(γ + δ) > 1 and c ≥ c * .…”

“…Meanwhile, they also proved that if R 0 ≤ 1 or c < c * , then (1.3) admits no traveling wave solutions. Their results [12] improved the corresponding results in [30]. Considering this biological fact that infected individuals often take time to move in space and are usually not in the same position at previous time, Wang et al [31] introduced a spatio-temporal delay term The kernel K(x − y, t − s) ≥ 0 depicts the interaction between the infected and susceptible individuals at location x and the present time t which occurred at location y and earlier time s, so the spatio-temporal delayed term K * I reflects that the infected individuals have the ability of movement and infectiousness during the latent period.…”

“…In the present paper, motivated by [12,40,47], we propose the spatio-temporal delayed counterpart of (1.6)…”

“…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.…”