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In this paper, we propose two new methods to solve large-scale systems of differential equations, which are based on the Krylov method. In the first one, the exact solution with the exponential projection technique of the matrix. In the second, we get a new problem of small size, by dropping the initial problem, and then we solve it in ways, such as the Rosenbrock and the BDF. Some theoretical results are presented such as an accurate expression of the remaining criteria. We give an expression of error report and numerical values to compare the two methods in terms of how long each method takes, and we also compare the approaches.

Rumor is an important form of social interaction, and its spreading has a significant impact on human lives. The optimal control theory is an important tool to better manage the spread of rumors. Most of the literature on rumor propagation models deals with quadratic cost functions relative to the control variable. In this paper, we have considered a time-delay rumor propagation model with saturated control functions and an objective function of $$L^1$$ -type linear with respect to the control variables. In the general case, the introduction of the delay in the dynamic systems represents the time lag between the action on the system and the response of the system to this action. The delay is incorporated in our model to make it more realistic and to describe the latency period. The existence of the optimal control pair is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.

In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic p(·)-Laplacian problem with Dirichlet-type boundary conditions and L 1 data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

In this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SIR epidemic model. The introduction of the delay in this model makes it more realistic and modelizes the latency period. In addition, the consideration of an SIR model with diffusion aims to better understand the impact of the spatial heterogeneity of the environment and the movement of individuals on the persistence and extinction of disease. First, we determined a threshold value $$R_0$$ of the delayed SIR model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that if $$R_01$$ , the disease-free equilibrium is unstable and there is a unique, asymptotically stable endemic equilibrium. Next, by constructing an appropriate Lyapunov function and using upper–lower solution method, we determine the threshold parameters which ensure the the global asymptotic stability of equilibria. Finally, we presented some numerical simulations to illustrate the theoretical results.

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is {-div[(1+uq)∇u]+u=γ∇u2/1+u1-q+f in Ω, u=0 on ∂Ω,}, where 00 and f belongs to some appropriate Lebesgue space. We give assumptions on f with respect to q and γ to show the existence and regularity results for the solutions of previous equation.

The model analyzed in this paper is based on the model set forth by [M.A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator–prey model with modified Leslie–Gower and Holling-type II schemes, Appl. Math. Lett. 16 (2003) 1069–1075, A.F. Nindjin, M.A. Aziz-Alaoui, M. Cadivel, Analysis of a a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., in Press.] with time delay, which describes the competition between predator and prey. This model incorporates a modified version of Leslie–Gower functional response as well as that of the Holling-type II. In this paper, we consider the model with one delay and a unique non-trivial equilibrium E * and the three others are trivial. Their dynamics are studied in terms of the local stability and of the description of the Hopf bifurcation at E * for small and large delays and at the third trivial equilibrium that is proven to exist as the delay (taken as a parameter of bifurcation) crosses some critical values. We illustrate these results by numerical simulations.

We propose a delayed SIR model with saturated incidence rate. The delay is incorporated into the model in order to model the latent period. The basic reproductive number R0 is obtained. Furthermore, using time delay as a bifurcation parameter, it is proven that there exists a critical value of delay for the stability of diseases prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. The model is extended to assess the impact of some control measures, by reformulating the model as an optimal control problem with vaccination and treatment. The existence of the optimal control is also proved. Finally, some numerical simulations are performed to verify the theoretical analysis.

We consider the haematopoietic stem cells model (HSC) with one delay introduced by Mackey [M.C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, blood 51 (1978) 5; M.C. Mackey, Mathematical models of haematopoietic cell replication and control, in: The Art of Mathematical Modelling: Case Studies in Ecology, Physiology and Biofluids, H.G. Othmer, F.R. Adler, M.A. Lewis, J.C. Dallon (Eds), Prentice-Hall, New York, 1997, p. 149] and Andersen and Mackey [L.K. Andersen, M.C. Mackey, Resonance in periodic chemotherapy: a case study of acute myelogenous leukemia, J. theor. Biol. 209 (2001) 113]. There are two possible stationary states in the model. One of them is trivial and the second E( *)(tau) depending on the delay is non-trivial . This paper investigates the stability of the non-trivial state and occurrence of the Hopf bifurcation depending on time delay. We prove the existence and uniqueness of a critical values tau(0) and tau of the delay such that E( *)(tau) is asymptotically stable for tautau(0) and unstable for tau(0)tautau . We show that E( *)(tau(0)) is a Hopf bifurcation critical point for an approachable model.

In epidemiological research literatures, a latent or incubation period can be medelled by incorporating it as a delay effect (delayed SIR models), or by introducing an exposed class (SEIR models). In this paper we propose a comparison of a delayed SIR model and its corresponding SEIR model in terms of local stability. Also some numerical simulations are given to illustrate the theoretical results.