Recently a new probability distribution, the spinned Poisson distribution (SPD), was introduced to model data collected in various health management situations, such as the spread of an infectious disease when infected cases are removed from the observed population. As the name suggests, the SPD is a generalization of the well known Poisson distribution. In this paper, simple data sets are used to show that there is no guarantee that the equations for the maximum likelihood estimates of the parameters associated with the SPD have a nontrivial solution. It follows that even for a simple data set where a nontrivial SPD may seem appropriate, the fitted SPD may be just a Poisson distribution.

In this paper, we obtain a collection of existence results of almost automorphic mild solutions to several stochastic functional differential equations in a real separable Hilbert space. The main technique is based upon some appropriate composition theorems combined with the fixed point method. Moreover, an example is also given to justify the practical usefulness of the established general theorems.

This paper is concerned with the abstract fractional differential equations with nonlocal condition. By using the contraction mapping principle and the theory of the measures of noncompactness and the condensing maps, we obtain the existence results of mild solutions for the above equations.

Psoriasis is an autoimmune prevalent chronic skin disease discriminated by T-Cells mediated hyperproliferation of epidermal Keratinocytes. In this research article, we extend the research work [1] and consider a mathematical model for Psoriasis, involving a set of differential equations with T-Cells, Dendritic Cells (DCs) and epidermal Keratinocytes. We introduce here the fractional-order differential equations into the mathematical model of Psoriasis with effect of Cytokines release to observe the impact of it on the cell-biological system. Analytical study on the basis of stability analysis with fractional derivative is furnished. Moreover numerical simulation through nonstandard finite difference methods has been applied for solving the fractional-order differential equations to support the analytical results. We have established that the effect of Cytokine network through exploring the suppressed memory, the inherited property of the system dynamics, in our mathematical model that contributes a greater impact for reducing the Keratinocyte cell population, causes the disease Psoriasis.

This paper presents a couple of existence results, related to the classes of functional equations of the form $x+k\ast x=f$, or $\frac{d}{dt}[\dot{x}+k\ast x]=f$, with $f, x\in AP_r(R, \,{\mathcal{C}})=$ the space of almost periodic functions defined by \[ AP_r(R,\, {\mathcal{C}})=\left\{f : f\simeq \sum_{j=1}^{\infty} f_j\,e^{i\lambda_j t},\,f_j\in {\mathcal{C}},\lambda_j\in R,\sum_{j=1}^{\infty}|f_j|^r \lt \infty\right\}, \] the norm being given by $|f|_r= \left(\sum_{j=1}^{\infty}|f_j|^r\right)^{\frac{1}{r}}$, for each $r\in [1, 2]$. The convolution product $k\ast x$, $k\in L^1(R,\, {\mathcal{C}})$, $x\in AP_r(R,\, {\mathcal{C}})$ is defined by \[ (k\ast x)(t)= \sum_{j=1}^{\infty} x_j\left( \int_R k(s)\,e^{\lambda_j\,s}\,ds\right)\,e^{i\lambda_j\,t}, \] where $x(t)\simeq \sum_{j=1}^{\infty} x_j\,e^{i\lambda_j\,t}$.

In this paper we introduce and study a new class of functions called $S^{p,q(x)}$pseudo-almost periodic (or weighted Stepanov-like pseudo-almost periodic functions with variable exponents), which generalizes the class of weighted Stepanov-like pseudo-almost periodic functions. Basic properties of these new spaces are established. The existence of weighted pseudo-almost periodic solutions to some first-order differential equations with $S^{p,q(x)}$pseudo-almost periodic coefficients will also be studied.

Harald Bohr was the founder of the theory of almost periodicity. The theory of almost periodic functions taking values in locally convex spaces was studied by G.M. N'Guérékata in his papers [9, 10]. Khan and Alsulami [12] studied the concept of almost periodicity in the general topological vector spaces. In this paper, we pursue their study further and extend the concept of weakly almost periodicity to topological vector spaces having non-trivial duals.

In this paper we study some existence, uniqueness, estimates and global asymptotic stability results for some functional integro-differential equations of fractional order with finite delay. To achieve our goals we make extensive use of some fixed point theorems as well as the so-called Pachpatte techniques.

The aim of this paper is to introduce of notion of weak variational solution in an abstract setting, although we are mainly interested in almost periodic type solutions. We give two existence and uniqueness theorems. Even if assumptions are strong, we obtain two theorems with explicit bounds.