Global Dynamics of a Spore Producing Pathogens Epidemic System with Nonlocal Diffusion Process

Abstract

In this paper, we analyze a nonlocal spatial epidemic model presenting the diffusion process of a spore producing plant pathogens responsible of one of the most destructive cocoa pods disease. The global existence, compactness and dissipativity of the semiflow generated by the system are established. By defining a threshold number (the basic reproduction number), we first express conditions for the existence of non-trivial stationary states. Next, we show that the qualitative dynamics of the model range from the extinction (i.e. the global stability of the disease-free stationary state), to the persistence (i.e. the global stability of the endemic stationary state) of the epidemics. Finally, we go through some simulations of our general analysis.

Mathematical analysis of a vaccination epidemic model with nonlocal diffusion

“…Infectious diseases have affected the human population and created a restricted standard of living. One of the best examples is the novel coronavirus (COVID- 19), which has influenced the public health sector, as well as social and economic systems in many countries (as of November 2022). Mathematical modeling is considered as one of the most effective manners of theoretically discussing the problem of infectious diseases, and numerous studies have been conducted on this topic (e.g., previous studies [1][2][3][4][5][6][7]).…”

“…Compared with the local diffusion represented by the Laplacian operator, nonlocal diffusion represents the movement of individuals not limited to one local area but extended to a wider area. Therefore, nonlocal diffusion models can address a larger class of natural phenomena and have recently attracted considerable attention (see Andreu-Vaillo et al [16] and other references [17][18][19][20][21][22][23][24][25][26][27][28][29]).…”

Growth bound and threshold dynamic for nonautonomous nondensely defined evolution problems

“…[f 2 ](t) := R N ∞ 0 K(• − y)r(a, y)e −µ 0 a f 2 (t − a, y)dady, ∀t ∈ R C 21 [f 1 ](t) := β(t, •)S(t) t −∞ e −δ(t−s) f 1 (s, •)ds, ∀t ∈ R.Using similar arguments as in[13], we conclude that sign(r(C) − 1) = sign(r(C 12 • C 21 ) − 1) where the linear operator C 12 • C 21 is given by(C 12 •C 21 )[f ](t) = y)r(a, y)e −µ 0 a β(t − a, y)S(t − a) t−a −∞ e −δ(t−a−s) f 1 (s, y)ds dady,for every f ∈ C p (R, Y ) and t ∈ R.…”

Growth bound and threshold dynamic for nonautonomous nondensely defined evolution problems

“…− y)r(a, y)e −µ 0 a f 2 (t − a, y)dady, ∀t ∈ RC 21 [f 1 ](t) := β(t, •)S(t) t −∞ e −δ(t−s) f 1 (s, •)ds, ∀t ∈ R.Using similar arguments as in[13], we conclude that sign(r(C) − 1) = sign(r(C 12 • C 21 ) − 1) where the linear operator C 12 • C 21 is given by(C 12 •C 21 )[f ](t) = R N ∞ 0 K(•−y)r(a, y)e −µ 0 a β(t − a, y)S(t − a) t−a −∞ e −δ(t−a−s) f 1 (s, y)ds dady, for every f ∈ C p (R, Y ) and t ∈ R. t, s, τ ) e − s s−τ ϑ(l−s+t,l,l−s+τ )dl w(t − τ, s − τ ) dsdτ 0 threshold dynamics of (4.25) is given by the spectral radius of the following linear operator (C[w](t))(a) = diag( S(t, a)) , s, τ )e − s s−τ ϑ(l−s+t,l,l−s+τ )dl w(t − τ, s − τ )dsdτ, for all t ∈ R, m ∈ C p (R, Y ) with the notation w(t)(a) = w(t, a). Setting w = w h w m the linear operator C takes the following formC[w] = C m [w m ] C h [w h ]where we have set for k = h, m(C k [w k ](t))(a) := Sk (t, a) ∞ 0 ∞ τ β k (t, s, τ )e − s s−τ ϑ k (l−s+t,l,l−s+τ )dl w k (t − τ, s − τ )dsdτ.Using similar arguments in[13], we deduce that r(C) − 1, r(C h • C m ) − 1 and r(C m • C h ) −1have the same sign. Note that r(C h • C m ) = r(C m • C h ) is actually the basic reproduction number of (4.25)…”