Traveling waves for a diffusive SIR-B epidemic model with multiple transmission pathways

In this work, we consider a diffusive SIR-B epidemic model with multiple transmission pathways and saturating incidence rates. We first present the explicit formula of the basic reproduction number R0. Then we show that if R0 > 1, there exists a constant c ∗ > 0 such that the system admits traveling wave solutions connecting the disease-free equilibrium and endemic equilibrium with speed c if and only if c ≥ c Since the system does not admit the comparison principle, we appeal to the standard Schauder’s fixed point theorem to prove the existence of traveling waves. Moreover, a suitable Lyapunov function is constructed to prove the upward convergence of traveling waves

Traveling waves for a diffusive SIR-B epidemic model with multiple transmission pathways

In this work, we consider a diffusive SIR-B epidemic model with multiple transmission pathways and saturating incidence rates. We first present the explicit formula of the basic reproduction number R0. Then we show that if R0 > 1, there exists a constant c ∗ > 0 such that the system admits traveling wave solutions connecting the disease-free equilibrium and endemic equilibrium with speed c if and only if c ≥ c Since the system does not admit the comparison principle, we appeal to the standard Schauder’s fixed point theorem to prove the existence of traveling waves. Moreover, a suitable Lyapunov function is constructed to prove the upward convergence of traveling waves

Optimal Control Applied to Population and Disease Models

This dissertation considers the use of optimal control theory in population models for the purpose of characterizing strategies of control which minimize an invasive or infected population with the least cost. Three different models and optimal control problems are presented. Each model describes population dynamics via a system of differential equations and includes the effects of one or more control methods. The first model is a system of two ordinary differential equations describing dynamics between a native population and an invasive population. Population growth terms are functions of the control, constructed so that the value of the control may affect each population differently. A novel existence result is presented for the case of quadratic growth functions. With parameters chosen to mimic the competition between cottonwood and salt cedar plants, optimal schedules of controlled ooding are displayed. The second model, a system of six ordinary differential equations, describes the spread of cholera in a human population through ingestion of Vibrio cholerae. Equations track movement of susceptible individuals to either an asymptomatic infected class or a symptomatic infected class through ingestion of bacteria, both in a hyperinfectious state and a less-infectious state. Recovered individuals temporarily move to an immune class before being placed back in the susceptible class. A new result quantities contributions to the basic reproductive number from multiple infectious classes. Within the model, three control functions represent rehydration and antibiotic treatment, vaccination, and sanitation. The cost-effective balance of multiple cholera intervention methods is compared for two endemic populations. The third model describes the spread of disease in both time and space using a system of three parabolic partial differential equations with convection-diffusion movement terms and no-flux boundary conditions. A control function representing vaccination is incorporated. State variables track the number of susceptible, infected, and immune individuals. Detailed analysis for the characterization of the optimal control is provided. The model and optimal control results are applied to the spread of rabies among raccoons with the control function determining the timing and placement of oral vaccine baits. Results illustrate cost-effective vaccine distribution strategies for both regular and irregular patterns of rabies propagation

Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods

We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit. The kinetic transport model is therefore considered within a spatial network, characterizing different places such as villages, cities, countries, etc. The transmission conditions in the nodes are analyzed and defined. Finally, the model is solved numerically on the network through a finite-volume IMEX method able to maintain the consistency with the diffusive limit without restrictions due to the scaling parameters. Several numerical tests for simple epidemic network structures are reported and confirm the ability of the model to correctly describe the spread of an epidemic

Kinetic Modelling of Epidemic Dynamics: Social Contacts, Control with Uncertain Data, and Multiscale Spatial Dynamics

In this survey we report some recent results in the mathematical modelling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker’s perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modelling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings

Hydrological, Anthropogenic and Ecological Processes in Cholera Dynamics

The present Thesis deals with understanding, measuring and modelling epidemic cholera. The relevance of the endeavour stems from the fact that mathematical epidemiology, properly guided by model-guided field validation, is a reliable and powerful tool to monitor and predict ongoing epidemics in time for action, and to save lives through evaluation of the effectiveness of mitigation policies or the deployment of medical staff and supplies. In recent years, waterborne diseases – and cholera in particular – have consolidated their role as a major threat for developing countries, where sanitation conditions are poor and the vulnerability to extreme events is highest. Several important features controlling the dynamics of occurrence and spreading of the disease in a region are studied in the present Thesis, from both theoretical and experimental perspective. Understanding the basic processes that regulate cholera infection is key to build reliable prediction tools. In this Thesis several driving mechanisms of cholera are investigated, with the objective of connecting together hydro-climatology, ecology and epidemiology in a comprehensive framework. First, the role of water volume fluctuations is analyzed partly through a bifurcation study in a mostly theoretical assessment. Such work is then particularized in a field campaign carried out in rural Bangladesh, where hydro-climatological variables and Vibrio cholerae concentrations have been monitored for more than a year in one of the ponds constituting the local water reservoir. Concomitantly, a procedure for the detection of Vibrio cholerae, based on flow cytometry, is tested in the field. Further emphasis is also given to the role of human mobility in disseminating the disease among different communities. In particular, the contribution of human mobility in the dispersal of vibrios along the hydrologic network is specifically analyzed. All the knowledge collected in these studies is then used to add essential details to a modeling framework that is applied to the dramatic case of the Haiti epidemic. A spatially explicit model, taking into account both hydrological transport and human mobility, is developed to simulate the spreading of the disease since its onset. It is also shown that the resurgence of the disease, coinciding with the rainy season of June-July 2011, can only be reproduced if hydrological forcings are considered. The framework is tested by forcing it with synthetic rainfall scenarios and projecting epidemiological outputs. It is shown that the model can quantify correctly the number of cases in a given time span, even when calibrated with limited information. This result allows then to use it as a tool to assess a priori the effectiveness of intervention policies, such as vaccination and sanitation. The effect of these two is tested both in the short- and in the long-term, with different results. Such endeavour represents the ultimate goal of the work presented in this Thesis – albeit further effort is needed to link together public health management and mathematical epidemiology in this field

Threshold Dynamics of a Stochastic S

A stochastic SIR model with vertical transmission and vaccination is proposed and investigated in this paper. The threshold dynamics are explored when the noise is small. The conditions for the extinction or persistence of infectious diseases are deduced. Our results show that large noise can lead to the extinction of infectious diseases which is conducive to epidemic diseases control

2014 Conference Abstracts: Annual Undergraduate Research Conference at the Interface of Biology and Mathematics

Conference schedule and abstract book for the Sixth Annual Undergraduate Research Conference at the Interface of Biology and Mathematics Date: November 1-2, 2014Plenary Speakers: Joseph Tien, Associate Professor of Mathematics at The Ohio State University; and Jeremy Smith, Governor\u27s Chair at the University of Tennessee and Director of the University of Tennessee/Oak Ridge National Lab Center for Molecular Biophysic