In this paper, we consider discrete growth–decay–fragmentation equations that describe the size distribution of clusters that can undergo splitting, growth and decay. The clusters can be for instance animal groups that can split but can also grow, or decrease in size due to birth or death of individuals in the group, or chemical particles where the growth and decay can be due to surface deposition or erosion. We prove that for a large class of such problems, the solution semigroup is analytic and compact and thus has the asynchronous exponential growth property; that is, the long-term behaviour of any solution is given by a scalar exponential function multiplied by a vector, called the stable population distribution, that is independent on the initial conditions.

The outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a public health emergency that turns the year 2020–2021 into annus horribilis for millions of people across international boundaries. The interspecies transmission of this zoonotic virus and mutated variants are aided by exposure dynamics of infected aerosols, fomites and intermediate reservoirs. The spike in the first, second and third waves of coronavirus confirms that herd immunity is not yet reached and everyone including livestock is still vulnerable to the infection. Of serious concern are the communitarian nature of agrarians in the livestock sector, aerogenous spread of the virus and attendant cytocidal effect in permissive cells following activation of pathogen recognition receptors, replication cycles, virulent mutations, seasonal spike in infection rates, flurry of reinfections and excess mortalities that can affect animal welfare and food security. As the capacity to either resist or be susceptible to infection is influenced by numerous factors, identifying coronavirus-associated variants and correlating exposure dynamics with viral aerosols, spirometry indices, comorbidities, susceptible blood types, cellular miRNA binding sites and multisystem inflammatory syndrome remains a challenge where the lethal zoonotic infections are prevalent in the livestock industry, being the hub of dairy, fur, meat and egg production. This review provides insights into the complexity of the disease burden and recommends precision smart-farming models for upscaling biosecurity measures and adoption of digitalised technologies (robotic drones) powered by multiparametric sensors and radio modem systems for real-time tracking of infectious strains in the agro-environment and managing the transition into the new-normal realities in the livestock industry.

The Lane-Emden equation, a nonlinear second-order ordinary differential equation, plays a fundamental role in theoretical physics and astrophysics, particularly in modeling the structure of stellar interiors. Also referred to as the polytropic differential equation, it describes the behavior of self-gravitating polytropic spheres. In this study, we present a novel approach to the solution of the eigenvalue problem which arises when considering the Lane-Emden equation for n = 0, 1, 2, 3, 4 using Physics-Informed Neural Networks (PINNs). The novelty of this work is that, we not only solve the Lane-Emden equation via PINNS but we also determine the eigenvalue, r, which is the stellar radius. Hyperparameter tuning was conducted using Bayesian optimization in the Optuna framework to identify optimal values for the number of hidden layers, number of neurons, activation function, optimizer, and learning rate for each value of n. The results show that, for n = 0, 1, PINNs achieve near-exact agreement with theoretical eigenvalues (errors < 0.000806%). While for more nonlinear cases, n = 2, 3 and n=4, PINNs yield errors below 0.0009% and 0.05% respectively, validating their robustness.

In this paper we study the discrete coagulation--fragmentation models with growth, decay and sedimentation. We demonstrate the existence and uniqueness of classical global solutions provided the linear processes are sufficiently strong. This paper extends several previous results both by considering a more general model and and also signnificantly weakening the assumptions. Theoretical conclusions are supported by numerical simulations.