arXiv subjects

Maia Martcheva

Publications and source records attributed to Maia Martcheva.

3 recordsLinked to original sources

arXiv2020

Solving Singular Control Problems in Mathematical Biology, Using PASA

In this paper, we will demonstrate how to use a nonlinear polyhedral constrained optimization solver called the Polyhedral Active Set Algorithm (PASA) for solving a general singular control problem. We present methods of discretizing a general optimal control problem that involves the use of the gradient of the Lagrangian for computing the gradient of the cost functional so that PASA can be applied. When a numerical solution contains artifacts that resemble "chattering'', a phenomenon where the control oscillates wildly along the singular region, we recommend a method of regularizing the singular control problem by adding a term to the cost functional that measures a scalar multiple of the total variation of the control, where the scalar is viewed as a tuning parameter. We then demonstrate PASA's performance on three singular control problems that give rise to different applications of mathematical biology. We also provide some exposition on the heuristics that we use in determining an appropriate size for the tuning parameter.

math.OC↗

arXiv2020

Modeling the effects of prosocial awareness on COVID-19 dynamics: A case study on Colombia

The ongoing COVID-19 pandemic has affected most of the countries on Earth. It has become a pandemic outbreak with more than 24 million confirmed infections and above 840 thousand deaths worldwide. In this study, we consider a mathematical model on COVID-19 transmission with the prosocial awareness effect. The proposed model can have four equilibrium states based on different parametric conditions. The local and global stability conditions for awareness free, disease-free equilibrium is studied. Using Lyapunov function theory and LaSalle Invariance Principle, the disease-free equilibrium is shown globally asymptotically stable under some parametric constraints. The existence of unique awareness free, endemic equilibrium and unique endemic equilibrium is presented. We calibrate our proposed model parameters to fit daily cases and deaths from Colombia. Sensitivity analysis indicates that the transmission rate and learning factor related to awareness of susceptibles are very crucial for reduction in disease related deaths. Finally, we assess the impact of prosocial awareness during the outbreak and compare this strategy with popular control measures. Results indicate that prosocial awareness has competitive potential to flatten the curve.

q-bio.PE↗

arXiv2013

Forward Hysteresis and Backward Bifurcation Caused by Culling in an Avian Influenza Model

The emerging threat of a human pandemic caused by the H5N1 avian influenza virus strain magnifies the need for controlling the incidence of H5N1 infection in domestic bird populations. Culling is one of the most widely used control measures and has proved effective for isolated outbreaks. However, the socio-economic impacts of mass culling, in the face of a disease which has become endemic in many regions of the world, can affect the implementation and success of culling as a control measure. We use mathematical modeling to understand the dynamics of avian influenza under different culling approaches. We incorporate culling into an SI model by considering the per capita culling rates to be general functions of the number of infected birds. Complex dynamics of the system, such as backward bifurcation and forward hysteresis, along with bi-stability, are detected and analyzed for two distinct culling scenarios. In these cases, employing other control measures temporarily can drastically change the dynamics of the solutions to a more favorable outcome for disease control.

q-bio.PE↗