Introduction to Dynamical Modeling Veronika Zarnitsyna, Department of Microbiology and Immunology, Emory University School of Medicine, [email protected] June 14th, 2018 kBI dI rBI gB(1-B/B ) max Bacteria, B Immune Response, I “All models are wrong, but some are useful”. George Box, 1978 Dynamical mechanistic models • Dynamical: Tracking how things change in time • Mechanistic: Having equations or computer rules that explicitly describe how things happen Within dynamical models, two broad classes are distinguished: • Deterministic: Assumption that the future is entirely predicted (determined) by the model • Stochastic: Assumption that random (stochastic) events affect the biological system, in which case a model can only predict the probability of various outcomes in the future Many types of Dynamical Models exist Compartmental « Agent-based Discrete time « Continuous time Deterministic « Stochastic The models we’ll be Homogeneous « Spatial focusing on, formulated Memory-less « With memory as Ordinary Differential Equations Small « Big Data-free « With data 4 Modeling The real world The conceptual world Observations Phenomena Model (analyses) Predictions Dym and Ivey 1980 5 Introduction to Dynamical models Veronika Zarnitsyna 6/14 Dynamical models are very common in biology as they provide insight into how various forces act to change a cell, an organism, a population, or an assemblage of species. Within dynamical models, two broad classes are distinguished: deterministic and stochastic. “Deterministic” models are based on the assumption that the future is entirely predicted (determined) by the model. “Stochastic” models are based on the assumption that random (stochastic) events affect the biological system, in which case a model can only predict the probability of variousMalthusian outcomes in the growth future. We willmodel focus on (1798 deterministic) models. 1. Simple exponential(a simple growth exponential model growth model) Historically, the first attempts to describe the biological processes were the population dynamics models. One of the famous model is Malthusian growth model (1798), sometimes called a simple exponential growth model. This model is essentially exponentialdiscrete-time growth bacterial based on agrowth constant model rate. Let ‘B’ be, for example, the size of a population of bacteria that can change with time. If we assume that the rate of growth of the population is proportionalbirth to the sizedeath of the population, it will give rise to the following equation: B = B +(bB dB )· = B + gB · t+· t t ≠ t t t dB gB = gB dt Bacteria, B where g is the growth rate of theTotal number of population. The solution to this is time step bacteria at the gt next time step B(tterm )=describes how change B(0)e happens (the mechanisms) You canThomas Robert Malthus (1766 verify that this is-1834) the solution.Total number of bacteria right now We can plot the population size B(t) as a function of g. Before you look at the plots, think of what you might expect. g=1 g=1 10000 g=0.3 g=0.3 g=0 1e+03 g=0 g=−1 g=−1 6000 1e+01 B(t) B(t) 01 − 1e 2000 03 0 − 1e 0 2 4 6 8 10 0 2 4 6 8 10 Time Time Notice: 1 Introduction to Dynamical models Veronika Zarnitsyna 6/14 Dynamical models are very common in biology as they provide insight into how various forces act to change a cell, an organism, a population, or an assemblage of species. Within dynamical models, two broad classes are distinguished: deterministic and stochastic. “Deterministic” models are based on the assumption that the future is entirely predicted (determined) by the model. “Stochastic” models are based on the assumption that random (stochastic) events affect the biological system, in which case a model can only predict the probability of various outcomes in the future. We will focus on deterministic models. 1. Simple exponential growth model Historically, the first attempts to describe the biological processes were the population dynamics models. One of the famous modelDiscrete is Malthusian- growthtime modelbacterial (1798), sometimes growth called model a simple exponential growth model. This model is essentially exponential growth based on a constant rate. Let ‘B’ be, for example, the size of a population of bacteria that can change with time. If we assume that the rate of growth of the population is proportionalbirth to the sizedeath of the population, it will give rise to the following equation: B = B +(bB dB )· = B + gB · t+· t t ≠ t t t dB = gB gB dt Bacteria, B where g is the growth rate of• theAssume population. b=12/hour, The solution d=2/hour, to this t=1 is hour. • B at start (t=0) B=( t100)=B(0)egt You can verify that this is the• What solution. do we get after 1,2,3,4,… hours? We can plot the population size B(t) as a function of g. Before you look at the plots, think of what you might expect. g=1 g=1 10000 g=0.3 g=0.3 g=0 1e+03 g=0 g=−1 g=−1 6000 1e+01 B(t) B(t) 01 − 1e 2000 03 0 − 1e 0 2 4 6 8 10 0 2 4 6 8 10 Time Time Notice: 1 Introduction to Dynamical models IntroductionIntroduction to DynamicalIntroduction to Dynamical modelsVeronika to models Dynamical Zarnitsyna models Veronika Zarnitsyna IntroductionVeronika toVeronika Dynamical Zarnitsyna Zarnitsyna models6/14 6/14 6/14 6/14 DynamicalVeronika models Zarnitsyna are very common in biology as they provide insight into how various forces act to change a cell, an organism, a population, or an assemblage of species. 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