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The aim of this work is to give sufficient conditions ensuring that the space PAP(𝕉, X, µ) of µ-pseudo almost periodic functions and the space PAA(𝕉, X, µ) of µ-pseudo almost automorphic functions are invariant by the convolution product f = k * f, k ∈ L 1(𝕉). These results establish sufficient assumptions on k and the measure µ. As a consequence, we investigate the existence and uniqueness of µ-pseudo almost periodic solutions and µ-pseudo almost automorphic solutions for some abstract integral equations, evolution equations and partial functional differential equations.

This work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). A multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism is proposed first. Indeed, there are human to human virus transmission, human to environment virus transmission, environment to human virus transmission and self-infection by susceptible individuals. The global stability of the disease-free equilibrium is shown when a given threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {T}}_{0} $$\end{document}T0 is less or equal to 1 and the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{0} $$\end{document}R0 is calculated. A convergence index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {T}}_{1} $$\end{document}T1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of infectious extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. Using Partial Rank Correlation Coefficient with a three levels fractional experimental design, the sensitivity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{0} $$\end{document}R0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {T}}_{0} $$\end{document}T0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {T}}_{1}$$\end{document}T1 to control parameters is evaluated. Following this study, the most significant parameter is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into account economic impacts of SARS-CoV-2, optimal fighting strategies are determined and discussed. The study is applied to real and available data from Cameroon with a model fitting. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy against SARS-CoV-2.

Cholera is a bacterial disease, its spread is strongly influenced by environmental factors and some socio-economic factors such as hygiene standards and nutrition of the population. This paper is devoted to the modelling of the impact of climatic factors and human behaviour on the spread of cholera. The mathematical modelling incorporates the direct transmission and the indirect transmission due to environmental knowledge. Taking into account the effect of the intra-annual variation of climatic factors on the transmission of cholera, a non-autonomous ordinary differential equations is proposed to describe the dynamics of the transmission of cholera. When the intra-annual variation of climate is not incorporated into the model, the latter becomes autonomous. The basic reproductive number is calculated and the stabilities of equilibria are investigated. In the non-autonomous case, the disease extinction and uniform persistence of disease are investigated. The results suggest that the transmission and spread of cholera can be affected by climatic factors, the proportion of the undernourished individuals and the proportion of people who respect the hygiene standards. Finally, some numerical simulations are proposed using the parameters values of climatic factors and socio-economic factors of some localities situated in Lake Chad border between Chad, Cameroon and Nigeria.

The aim of this work is to establish several results on the existence and regularity of solutions for some nondensely nonautonomous partial functional differential equations with finite delay in a Banach space. We assume that the linear part is not necessarily densely defined and generates an evolution family under the conditions introduced by N. Tanaka.We show the local existence of the mild solutions which may blow up at the finite time. Secondly,we give sufficient conditions ensuring the existence of the strict solutions. Finally, we consider a reaction diffusion equation with delay to illustrate the obtained results.

Cholera is a bacterial disease caused by the bacterium Vibrio cholerae that requires optimal temperature and environmental conditions to survive. It is well known that climate change, influence of ecology, flood and droughts can affect the concentration of the bacterium in the environment. The goal of this article is to establish the effects of hygiene, famine, climate and environment on the transmission and spread of cholera. The transmission dynamics of the disease are modeled with a non-autonomous system of ordinary differential equations that is coupled to a model of intra-annual variation of Vibrio cholerae in the environment. When the intra-annual variation of Vibrio cholerae is not incorporated into the model, this latter becomes autonomous and we then give an explicit formulation of the basic reproductive number. In the non-autonomous case, we make analytically explicit two thresholds that allow to exhibit cases where disease extinction otherwise disease uniform persistence may occur. Finally, some numerical simulations allow to study the evolution of the cholera spread according the different environmental factors.

The aim of this work is to study the existence of a periodic solution for some nondensely nonautonomous partial functional differential equations with infinite delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. We use Massera’s approach (Duke Math 17:457–475, 1950), we prove that the existence of a bounded solution on \(\mathbb {R}^{+}\) implies the existence of an \(\omega \)-periodic solution. In nonlinear case, we use a fixed point for multivalued maps to show the existence of a periodic solution. Finally, we consider a reaction diffusion equation with delay to illustrate the main results of this work.

In this paper we propose two nonlinear models for the control of anthracnose disease. The first is an ordinary differential equation (ODE) model which represents the within-host evolution of the disease. The second includes spatial diffusion of the disease in a bounded domain. We demonstrate the well-posedness of those models by verifying the existence of solutions for given initial conditions and positive invariance of the positive cone. By considering a quadratic cost functional and applying a maximum principle, we construct a feedback optimal control for the ODE model which is evaluated through numerical simulations with the scientific software Scilab. For the diffusion model we establish under some conditions the existence of an optimal control with respect to a generalized version of the cost functional mentioned above. We also provide a characterization for this optimal control., Biomath 3 (2014)

Climate change influences more and more of our activities. These changes led to environmental changes which has in turn affected the spatial and temporal distribution of the incidence of vector-borne diseases. To establish the impact of climate on contact rate of vector-borne diseases, we examine the variation of prevalence of diseases according to season. In this paper, the goal is to establish that the basic reproductive number [Formula: see text] depends on the duration of transmission period and the date of the first conta-mination case that was declared ([Formula: see text]) in the specific case of malaria. We described the dynamics of transmission of malaria by using non-autonomous differential equations. We analyzed the stability of endemic equilibrium (EE) and disease-free equilibrium (DFE). We prove that the persistence of disease depends on minimum and maximum values of contact rate of vector-borne diseases.

We give various equivalent formulations to the (partially) open problem about $L^p$-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For $p\geq 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we give in addition new necessary conditions on the number of derivatives required in its definition.

An affirmative answer is given to the following conjecture of R. Coifman and R. Rochberg: in any symmetric Siegel domain of type II, the dual of the Bergman space A 1 {A^1} coincides with the Bloch space of holomorphic functions and can be realized as the Bergman projection of L ∞ {L^\infty } .

We determine a defining kernel for the Bergman projection of L ∞ {L^\infty } in tubes over cones of real, symmetric, positive-definite matrices.

We prove that the Bloch space coincides with the space BMOA in the tube over the spherical cone of R3; this extends a well-known onedimensional result. Introduction. Let Q be a symmetric Siegel domain of type II contained in Cn. Let V denote the Lebesgue measure in Q and H(Q) the space of holomorphic (or analytic) functions in Q. When n = 1 and Q = 7r+ = {z E C: Imz > 0}, a Bloch function is an element f of H(7r+) which satisfies the estimate llf l = sup {fyf'(Z)I} < o. z=x+iyE7r+ The Bloch space 5 of ir+ is then the quotient space of the space of Bloch functions by the subspace of constant functions. It is well known that in r+, the Bloch space 5 coincides with the quotient space BMOA of the space of BMO analytic functions by the subspace of constant functions. The definition of BMO in r+ is the same as that of (solid) BMO in the unit disk (cf. [6, p. 631]): in ir+, a locally integrable function f is said to be BMO if there exists a constant C such that for any disk D contained in 7r+, there is a constant fD such that 1D ID f -fD| dV < C. DuD In C2, this result can easily be extended to the cartesian product (7r+)2 of two upper half-planes. In this case, a Bloch function is an element of H[(ir+)2] which satisfies the estimate t92 lIf lw = sup jYoYi gz0 z1 f(z) < 00. z=(zo,zl)=(xo+iyoxl+iyl)E(7r+)2 oZ The Bloch space 5 of (wr+)2 is then the quotient space of the space of Bloch functions by the subspace = {f E H[(7r+ )2]: t,a f(z) ?} Received by the editors May 1, 1986 and, in revised form, December 4, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 32M15, 46E99, 47B38.