Mathematical Modeling and Simulation with MATLAB

TextbookThis textbook attempts to provide you with an overview of the commonly used basic mathematical models, as well as a wide range of applications. It offers a perspective that brings you back to the modeling process and the assumptions that go into it.University of Alaska Southeast Viterbo UniversityPreface 5 1 Introduction 7 1.1 What is Modeling? 1.1.1 Types of Mathematical Models 1.2 Modeling with Equations 1.3 Modeling with Recurrence Relations 1.4 Modeling with Di erential Equations 1.5 Stochastic Models 1.6 Exercises 2 Programming in MATLAB 17 2.1 Why is Programming Important? 2.2 MATLAB Basics 2.3 MATLAB Functions and Terminology 2.4 Plotting Points and Curves 2.4.1 Using ezplot() 2.4.2 Using plot() 2.5 The Symbolic Toolkit 2.6 For Loops 2.7 While Loops 2.8 Conditional Statements 2.9 Exercises 3 Iterative Methods 35 3.1 Fixed Point Iteration 3.2 MATLAB function les 3.3 The Bisection Method 3.4 Newton's Method 3.5 Exercises CONTENTS 3 4 Matrices 46 4.1 Matrix Review 4.2 Eigenvectors and Eigenvalues 4.3 Summary of MATLAB commands 4.4 Exercises 5 Discrete Models 60 5.1 Introduction 5.2 Linear Dynamical Systems 5.3 State, Age, and Stage Matrix Models 5.4 Markov Chains 5.5 Higher Order and Nonlinear Discrete Dynamical Systems 5.6 Exercises 6 Continuous Models 81 6.1 Modeling with Di erential Equations 6.1.1 Euler's Method 6.1.2 Improved Euler's Method 6.1.3 Qualitative Analysis of Differential Equations 6.2 Systems of Differential Equations 6.3 Qualitative Analysis of Systems of Differential Equations 6.4 Linear systems 6.5 Nonlinear Systems of Equations 6.6 Exercises 7 Stochastic Modeling 113 7.1 Discrete Random Variables 7.2 Continuous Random Variables 7.3 Properties of Random Variables 7.4 Monte Carlo Integration 7.5 The Binomial Distribution 7.6 The Normal Distribution 7.7 Waiting Times 7.7.1 The Poisson Distribution 7.7.2 The Exponential Distribution 7.7.3 Sampling from the Exponential and Poisson Distributions 7.8 Stochastic Processes 7.8.1 Poisson Processes 7.8.2 Birth Death Processes 7.9 Exercises A In-class Activities and Exercises 145 CONTENTS 4 B Solutions to in-class Exercises 188 C Selected Solutions 193 D MATLAB Commands 198 E Debugging 202 F Completed Proofs 204 Index 20

Analysis of the sensitivity properties of a model of vector-borne bubonic plague

Model sensitivity is a key to evaluation of mathematical models in ecology and evolution, especially in complex models with numerous parameters. In this paper, we use some recently developed methods for sensitivity analysis to study the parameter sensitivity of a model of vector-borne bubonic plague in a rodent population proposed by Keeling & Gilligan. The new sensitivity tools are based on a variational analysis involving the adjoint equation. The new approach provides a relatively inexpensive way to obtain derivative information about model output with respect to parameters. We use this approach to determine the sensitivity of a quantity of interest (the force of infection from rats and their fleas to humans) to various model parameters, determine a region over which linearization at a specific parameter reference point is valid, develop a global picture of the output surface, and search for maxima and minima in a given region in the parameter space

A Replication-Competent Adenovirus-Human Immunodeficiency Virus (Ad-HIV) tat and Ad-HIV env Priming/Tat and Envelope Protein Boosting Regimen Elicits Enhanced Protective Efficacy against Simian/Human Immunodeficiency Virus SHIV(89.6P) Challenge in Rhesus Macaques

We previously demonstrated that replication-competent adenovirus (Ad)-simian immunodeficiency virus (SIV) recombinant prime/protein boost regimens elicit potent immunogenicity and strong, durable protection of rhesus macaques against SIV(mac251). Additionally, native Tat vaccines have conferred strong protection against simian/human immunodeficiency virus SHIV(89.6P) challenge of cynomolgus monkeys, while native, inactivated, or vectored Tat vaccines have failed to elicit similar protective efficacy in rhesus macaques. Here we asked if priming rhesus macaques with replicating Ad-human immunodeficiency virus (HIV) tat and boosting with the Tat protein would elicit protection against SHIV(89.6P). We also evaluated a Tat/Env regimen, adding an Ad-HIV env recombinant and envelope protein boost to test whether envelope antibodies would augment acute-phase protection. Further, expecting cellular immunity to enhance chronic viremia control, we tested a multigenic group: Ad-HIV tat, -HIV env, -SIV gag, and -SIV nef recombinants and Tat, Env, and Nef proteins. All regimens were immunogenic. A hierarchy was observed in enzyme-linked immunospot responses (with the strongest response for Env, followed by Gag, followed by Nef, followed by Tat) and antibody titers (with the highest titer for Env, followed by Tat, followed by Nef, followed by Gag). Following intravenous SHIV(89.6P) challenge, all macaques became infected. Compared to controls, no protection was seen in the Tat-only group, confirming previous reports for rhesus macaques. However, the multigenic group blunted acute viremia by approximately 1 log (P = 0.017), and both the multigenic and Tat/Env groups reduced chronic viremia by 3 and 4 logs, respectively, compared to controls (multigenic, P = 0.0003; Tat/Env, P < 0.0001). The strikingly greater reduction in the Tat/Env group than in the multigenic group (P = 0.014) was correlated with Tat and Env binding antibodies. Since prechallenge anti-Env antibodies lacked SHIV(89.6P)-neutralizing activity, other functional anti-Env and anti-Tat activities are under investigation, as is a possible synergy between the Tat and Env immunogens