Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

Abstract

We consider an age-dependent s-i-s epidemic model with diffusion whose mortality is unbounded. We approximate the solution using Galerkin methods in the space variable combined with backward Euler along the characteristic direction in the age and time variables. It is proven that the scheme is stable and convergent in optimal rate in l ∞,2 (L 2 ) norm. To investigate the global behavior of the discrete solution resulting from the algorithm, we reformulate the resulting system into a monotone form. Positivity of the nonlocal birth process is proved using the positivity of the first eigenvalue of the resulting matrix system and using the fact that the positivity is preserved along the characteristics. The difference equation of the steady state coupled with nonlocal birth process is solved by developing monotone iterative schemes. The stability of the discrete solution of the steady state is then analyzed by constructing suitable positive subsolutions.

Mathematical analysis for an epidemic model with spatial and age structure

“…[4,7,15,17,18]. If diffusion is taken into consideration epidemic models with age structure are studied in [5], where existence of periodic solutions is investigated in the case V = R n and with a time periodic forcing, in [12] and [13], where existence of traveling wave solutions is investigated in the scalar case and in [19], where approximate solutions using Galerkin methods are considered. More related to our model are the papers [10] and [16], where similar SIR models, with spatial and age structure, are studied in a L 2 setting or a C setting and under different assumptions.…”

Stability and bionomic analysis of fuzzy parameter based prey–predator harvesting model using UFM

“…For the development of human society, it is undoubtedly essential problem how to utilize the biological resources. In the last decades, ordinary differential equations is used to present this type of problem and provide some mathematical answer and explanations [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. However, it is notable that the biological parameters used in the differential equations are not always fixed.…”

Global behavior of SIS epidemic models with age structure and spatial heterogeneity

“…They proved that if a nontrivial endemic equilibrium (or a periodic solution) exists, then it is globally stable; however, the threshold condition for the existence of such an endemic equilibrium was not obtained. In [21], Kim studied an SIS epidemic model with an age structure and spatial diffusion. The model was discretized using the Galerkin method and the backward Euler method, and the nontrivial endemic equilibrium was shown to be globally asymptotically stable when it exists.…”