The replication-transmission relativity theory, currently used to inform the development of multiscale models of infectious disease dynamics, needs a revision and extension to accommodate new basic science and clinical information about infectious disease dynamics. In this article, we revise and extend the replication-transmission relativity theory into a new scientific theory of infectious disease dynamics called the universal theory for the multiscale modelling of infectious disease dynamics. This new theory states that, for every host–pathogen interaction that results in an infectious disease system, there is no privileged or absolute scale of a disease system form that would determine the dynamics of the infectious disease system, only interactions between the scales of a level of organisation of the pathogen-centred disease system form and the scales of the corresponding levels of organisation of the host-centred disease system form. We further explain the utility of this theory, which is reflected in its flexibility and ability to incorporate new information and explain previous information that could not be accounted for by the replication-transmission relativity theory of infectious disease dynamics.

This study focuses on the use of Physics-Informed Neural Networks (PINNs) to solve the 1D Advection–Diffusion–Reaction (ADR) equation. The performance of the PINN model is evaluated in comparison with the classical Crank–Nicolson Finite Difference Method (CNFDM) and validated against analytical solutions to assess improvements in accuracy, robustness, and flexibility. Quantitative analysis reveals that the PINN achieved a high level of accuracy with absolute errors ranging from approximately $2.13\times {10}^{-4}$ to $1.17\times {10}^{-3}$ across the spatial domain. The study utilizes a neural network architecture with two hidden layers of 80 neurons each, optimized through a two-stage training process involving Adam and L-BFGS optimizers. This work contributes to the growing field of physics-informed machine learning by demonstrating the strengths and quantitative reliability of the PINN technique for solving complex partial differential equations in transport phenomena.

Optimal control problems are a class of optimisation problems with partial differential equations as constraints. These problems arise in many application areas of science and engineering. The finite element method was used to transform the optimal control problems of an elliptic partial differential equation into a system of linear equations of saddle point form. The main focus of this paper is to characterise and exploit the structure of the coefficient matrix of the saddle point system to build an efficient numerical process. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The numerical solution of saddle point problems is a computational task since well known numerical schemes perform poorly if they are not properly preconditioned. The main task of this paper is to construct a preconditioner the mimic the structure of the system coefficient matrix to accelerate the convergence of the generalised minimal residual method. Explicit expression of the eigenvalue and eigenvectors for the preconditioned matrix are derived. The main outcome is to achieve optimal convergence results in a small number of iterations with respect to the decreasing mesh size $h$ and the changes in $δ$ the regularisation problem parameters. The numerical results demonstrate the effectiveness and performance of the proposed preconditioner compared to the other existing preconditioners and confirm theoretical results. Partial differential equations (PDEs), PDE-optimal control problems, saddle point problem, block preconditioners, preconditioned generalised minimal residual method (PGMRES). 49J20, 65M22, 65M55, 65M60 COPYRIGHT: © Global Science Press

This paper investigates the optimal cash crop combination at a small-scale cash crop farm in Vhembe District, Limpopo Province, South Africa in which green maize, cabbages, tomatoes, spinach, mustard, butternut and sweet potatoes are grown. To get optimum farm outputs, decisions on crop combination and operational activities in crop production are crucial. Proper farm planning and resource allocation play a significant role in optimising farm revenues. It was observed that the farmer used traditional methods of allocating resources, which lead to a less profitable crop mix. In view of this, in this study, linear programming model was formulated using data collected from a farm concerning the past crop combinations and allocation of resources in crop planning and production to determine the best crop combination that maximizes net income given limited resources such as land, labour, capital, and others. The simplex method of linear programming works by first locating a feasible solution and then relocating to any vertex of the feasible set that improves the cost function. Eventually, a point is reached beyond which no further movement improves the cost function. The results of the developed linear programming model were compared to past farming practices based on experience, leading to the conclusion that crops and limited resources were not optimally allocated. The results clearly demonstrate the optimal crop combination and allocation of scarce resources that the farm could have considered to yield maximum returns. It is observed that the proposed linear programming model is appropriate for finding the optimal land allocation criteria for the cash crops in the study area. The optimal crop mix from the linear programming two phase simplex method show that the farmer should grow the following crop mix, 1.16 ha of green mealies, 2.64 ha of cabbages, 0.8 ha of tomatoes, 1.2 ha of mustard, 0.4 ha spinach, 0.4 ha of butternut and 0.4 ha of sweet potatoes with the gross income of R 740 800. The linear programming model resulted in a 37.8 percent increase in profit margin. Based on the results obtained from this study, it is recommended that the small-scale cash crop farmer should invest more in producing crops that give maximum profit. MATLAB software was used to determine the optimal values of the decision variables. Key words: Cash crops allocation, crop combination, net income, linear programming (LP), resource allocation.

In this study, the optimal control problem is considered. This is an important class of partial differential equations constrained optimization problems. The constraint here is an elliptic partial differential equation with Neumann boundary conditions. The discretization of the optimality system produces a block coupled algebraic system of equations of saddle point form. The solution of such systems is a major computational task since they require specialized methods. Constructing robust, fast and efficient solvers for their numerical solution has preoccupied the computational science community for decades and various approaches have been developed. The approaches involve solving simultaneously for all the unknowns using a coupled block system, the segregated approach where a reduced system is solved and the approach of reducing to a fixed point form. Here the minimum residual solver with ideal preconditioning is applied to the unreduced 3 by 3 and reduced 2 by 2 coupled systems and compared to the multigrid method applied to the compact fixed-point form. The two methods are compared numerically in terms of iterative counts and computational times. The numerical results indicate that the two methods produce similar outcomes and the multigrid solver becoming very competitive in terms of the iterative counts though slower than preconditioned minimum residual solver in terms of computational times. For all the approaches, the two methods exhibited mesh and parameter independent convergence. The optimal performance of the two methods is verified computationally and theoretically.