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Mathematical models are widely used to describe and control the dissemination of diseases in the population. Stochasticity is introduced to bring new insight into the modelling of population dynamics of diseases. Many systems in nature divulge stochasticity and are subject to stochastic perturbations. In this talk, we present differential equations with stochastic perturbations. We present briefly the existence and uniqueness of the global positive solution to the stochastic system. We present the stability conditions and illustrate the simulation results for the proposed model. The model parameters are estimated using the updated data estimation method, and the proposed model is validated. Finally, we present a computational experiment and validate the model by performing statistical analysis for the actual data for some diseases like COVID-19 and acute respiratory diseases.

We consider a single server retrial queue (RTQ) with no waiting space at the service station, A finite number M of orbits, with the first M-1 having finite capacities K, 2K, ...., (M-1)K in the increasing order, and the Mth one having infinite capacity are provided for retrial customers, depending on the number of retrials they have completed-- orbit one seats tho primary customers who encounter a busy server on their arrival, ... the (M-1)th accommodates those who have completed M-2 retrials and the last one is for those who is offered either the last retrial/direct access to the server through a SEARCH by the service system at the EPOCH OF COMPLETION OF A SERVICE. This mechanism provides exact information of the number of retrials an arbitrarily chosen customer in the system has completed. The increasing nature of the capacities of the orbits together with the assumption that retrial rates from orbits increase in a sequential manner, ensure minimum loss of retrial customers. This system is analysed under the assumption that the primary arrivals follow a Poisson process; inter-retrial times in each orbit are exponentially distributed. Customers in the last orbit are endowed with a privilege of being either taken for service at a customer service completion epoch through a SEARCH MECHANISM requiring negligible time or through the retrial. Two cases are considered: 1. Retrial customers return to orbit with probability one; 2. such customers abandon the system with probability p. A comparison of the performance of these two is carried out by introducing separate cost functions.

Often in several fields of real world applications future developments are functions of past events. Hence, the processes describing these situations cannot be Markovian. But a Hawkes process has an intensity that is stochastic and depends on the past history of the process itself, while an inverse Hawkes process is a process that has constant intensity and stochastic jump size, depending on its past history. That is why, this paper considers processes that are obtained by combining Hawkes processes and inverse Hawkes processes. In this context the asymptotic results studied are generalized versions of the Law of Large Numbers and of the Central Limit Theorem.

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Reliability is defined to be the ability of a system or component to perform its required functions under stated conditions for a specified period of time. In the world of broadband network, different types of communication networks are facing a large demand due to its high- speed wireless data transmission, extensive coverage from base stations, and a luxury of upgrading protocol software. In this talk, an analytical model is discussed to determine reliability attributes of third generation and beyond Universal Mobile Telecommunication Systems (UMTS), Voice over Internet Protocol (VoIP), Long-Term Evolution Advanced (LTE-A), Vehicular Ad hoc Network (VANET), High-Altitude Platform (HAP) networks. These networks are modeled using stochastic models such as Markov chains, semi-Markov process, reliability block diagrams and Markov reward models to obtain these attributes. Numerical results illustrate the applicability of the proposed analytical model. In addition, it can help the guarantee of network connectivity after any failure, without over dimensioning the networks.

It is not well known that a nonsingular matrix with the so-called triangle property has a tridiagonal inverse. In this talk, I will give an overview of this matrix class. I will also touch upon some very recent results that we have obtained, for a related class of matrices whose inverses are (what we refer to as) bidiagonal matrices with the south-west structure.

Boundary value problems for PDEs employ subspace of Lp spaces. Boundaries of open sets have measure zero. Hence restriction of a function to a boundary is not well defined. Fractional order Sobolev spaces provide the required functional spaces using which Trace maps can be defined. They give the appropriate generalization of restriction operators.

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