MA 777: Topics in Mathematical Biology

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MA 777: Topics in Mathematical Biology David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma777/ Spring 2018 David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 1 / 15

Figure: from the Society for Industrial and Applied Mathematics (SIAM). David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 2 / 15

What is a Mathematical Model? A mathematical model is a description of a system using mathematical concepts and language. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 3 / 15

What is a Mathematical Model? A mathematical model is a description of a system using mathematical concepts and language. Modeling is a process that uses math to represent, analyze, make predictions, or otherwise provide insight into real world phenomena. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 3 / 15

What is a Mathematical Model? A mathematical model is a description of a system using mathematical concepts and language. Modeling is a process that uses math to represent, analyze, make predictions, or otherwise provide insight into real world phenomena. The process of developing a mathematical model is termed mathematical modeling. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 3 / 15

What is a Mathematical Model? A mathematical model is a description of a system using mathematical concepts and language. Modeling is a process that uses math to represent, analyze, make predictions, or otherwise provide insight into real world phenomena. The process of developing a mathematical model is termed mathematical modeling. Mathematical modeling is used in Physics Biology Economy etc. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 3 / 15

Types of Mathematical Models Many dynamic modeling approaches have been used over the last 6-7 decades for modeling biological systems. As a result, a large variety of models exists today. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 4 / 15

Types of Mathematical Models Many dynamic modeling approaches have been used over the last 6-7 decades for modeling biological systems. As a result, a large variety of models exists today. Generally, dynamic models can be classified according to how the time and the population of gene products are treated. Thus, there exist models based on continuous populations and continuous time such as systems of ordinary differential equations, David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 4 / 15

Types of Mathematical Models Many dynamic modeling approaches have been used over the last 6-7 decades for modeling biological systems. As a result, a large variety of models exists today. Generally, dynamic models can be classified according to how the time and the population of gene products are treated. Thus, there exist models based on continuous populations and continuous time such as systems of ordinary differential equations, models based on discrete populations and continuous time such as models based on the Gillespie formulation and their generalizations, and David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 4 / 15

Types of Mathematical Models Many dynamic modeling approaches have been used over the last 6-7 decades for modeling biological systems. As a result, a large variety of models exists today. Generally, dynamic models can be classified according to how the time and the population of gene products are treated. Thus, there exist models based on continuous populations and continuous time such as systems of ordinary differential equations, models based on discrete populations and continuous time such as models based on the Gillespie formulation and their generalizations, and models based on discrete population and discrete time frameworks such as Boolean Networks (BNs) and their stochastic variants. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 4 / 15

Other types of models 1 Statistical models. 2 Agent based models. 3 Animal models. 4 Logical models. 5 Petri nets. 6 PDEs and stochastic PDEs. 7 etc. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 5 / 15

Deterministic vs Stochastic { 1. All parameters are fixed, Deterministic 2. Dynamics is fixed. 1. Even when all parameters are know, the system may follow different trajectories. Stochastic 2. Although the final state might not be determined exactly, there are some parameters that govern the outputs. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 6 / 15

Variability during phage lambda infection Figure: A. Arkin, J. Ross, and H. H. McAdams. Genetics, 149(4), 1998. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 7 / 15

Modeling framework trade-off Desirable Requirements Accuracy Feasibility Fine model high high low Coarse model low "medium" high Requirements Accuracy Feasibility Model medium high high David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 8 / 15

Simulating Dynamics of Chemically Reacting Systems Data requirement Master Equa+on Intractable either numerically or analy+cally. Accurate but very slow. Con+nuous Gillespie Algorithm Sta+s+cally correct but s+ll slow Simple procedure Discrete Stochas+c ODE Determinis+c plus a stochas+c part Kine+cs needed Stochas+c DDS Stochas+c No kine+cs needed but some probability parameters are needed ODEs Determinis+c Kine+cs needed Discrete Dynamical Systems Determinis+c No kine+cs needed David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 9 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. Bifurcation analysis. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. Bifurcation analysis. Sensitivity analysis. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. Bifurcation analysis. Sensitivity analysis. Poincare-Bendixson theorem. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. Bifurcation analysis. Sensitivity analysis. Poincare-Bendixson theorem. Gillespie simulations. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. Bifurcation analysis. Sensitivity analysis. Poincare-Bendixson theorem. Gillespie simulations. Boolean networks. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. Bifurcation analysis. Sensitivity analysis. Poincare-Bendixson theorem. Gillespie simulations. Boolean networks. Markov chains. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. Bifurcation analysis. Sensitivity analysis. Poincare-Bendixson theorem. Gillespie simulations. Boolean networks. Markov chains. Perron-Frobenius theorem. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

How much you know about math modeling? Have you seen the following topics before? Dynamical systems. Bifurcation analysis. Sensitivity analysis. Poincare-Bendixson theorem. Gillespie simulations. Boolean networks. Markov chains. Perron-Frobenius theorem. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 10 / 15

Modeling population growth for single species Let N(t) be the population of a species at time t, then the rate of change can be modeled by dn dt = births deaths + migration David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 11 / 15

Modeling population growth for single species Let N(t) be the population of a species at time t, then the rate of change can be modeled by dn dt = births deaths + migration Example (Exponential growth) The simplest model has no migration and the birth and death are proportional to N, dn dt = bn dn N(t) = N 0 e (b d)t where b and d are positive constant and the initial population N(0) = N 0. This approach, due to Malthus in 1798, is fairly unrealistic. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 11 / 15

Modeling population growth for single species Example (Logistic growth) Verhulst (1838-1845) proposed that a self-limiting process should operate when a population becomes too large. He proposed ( dn = rn 1 N ) ( 1 dn or = r 1 N ) dt K N dt K where r and K are positive constants. In this model, the per capita growth rate decreases linearly with population size, r(1 N/K ). David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 12 / 15

Modeling population growth for single species Example (Logistic growth) Verhulst (1838-1845) proposed that a self-limiting process should operate when a population becomes too large. He proposed ( dn = rn 1 N ) ( 1 dn or = r 1 N ) dt K N dt K where r and K are positive constants. In this model, the per capita growth rate decreases linearly with population size, r(1 N/K ). If N(0) = N 0, the solution is N(t) = N 0 Ke rt K + N 0 (e rt 1) K as t The constant K is the carrying capacity. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 12 / 15

Modeling population growth for single species Figure: Logistic growth. Notice the qualitative difference when N 0 < K /2 and K /2 < N 0 < K. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 13 / 15

Modeling population growth for single species Example In general, we will consider dn dt = f (N) where f (N) is a nonlinear function. The equilibrium points N are the solutions of f (N) = 0. N is locally stable if f (N ) < 0. N is unstable if f (N ) > 0. Proof. Let N(t) = N + n(t). Show that n(t) = n(0)e rt, where r = f (N ). David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 14 / 15

The year of Mathematical Biology 2018 is a joint venture of the European Mathematical Society (EMS) and the European Society for Mathematical and Theoretical Biology (ESMTB). For more information, see http://euro-math-soc.eu/year-mathematical-biology-2018. David Murrugarra (University of Kentucky) Lecture 1: Mathematical Models MA 777 Spring 2018 15 / 15

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