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Introduction to Dynamical Modeling

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Introduction to Dynamical Modeling 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. Within dynamical models, two broad classes Dynamical models are6/14 very common in biology as they provide insight into how various forces act to change DynamicalDynamical models models are very are common very commonare in distinguished: biology in biology as they deterministic as provide they provide insight and insight stochastic. into how into various “Deterministic”how various forces forcesact models to act change to are change based on the assumption that the a cell, an organism, a population, or an assemblage of species. Within dynamical models, two broad classes a cell,a an cell, organism, an organism, a population, a population, or an assemblage or an assemblage of species. of species. Within Within dynamical dynamical models, models, two broad two broad classes classes arefuture distinguished: is entirely deterministic predicted (determined) and stochastic. by “Deterministic” the model. “Stochastic” models are models based on are the based assumption on the assumption that the are distinguished:are distinguished: deterministic deterministic and stochastic. and stochastic. “Deterministic” “Deterministic” models models are based are on based the on assumption the assumption that the that the Dynamical models are very commonfuturethat in is biology random entirely as (stochastic) predicted they provide (determined) events insight aff intoect by the how the biological various model. forces “Stochastic” system, act in to which change models case are a based model on can the only assumption predict the futurefuture is entirely is entirely predicted predicted (determined) (determined) by the by model. the model. “Stochastic” “Stochastic” models models are based are based on the on assumption the assumption a cell, an organism, a population,that orprobability random an assemblage (stochastic) of various of species. outcomes events Within aff inect the dynamical the future. biological We models, will system, focus two broad in on which deterministic classes case a model models. can only predict the that randomthat random (stochastic) (stochastic) events events affect a theffect biological the biological system, system, in which in which case a case model a model can only can predict only predict the the are distinguished: deterministicprobability and stochastic. of various “Deterministic” outcomes models in the future. are based We on will the focus assumption on deterministic that the models. probabilityprobability of various of various outcomes outcomes in1. the Simple in future. the future. exponential We will We focus will growth on focus deterministic on model deterministic models. models. future is entirely predicted (determined) by the model. “Stochastic” models are based on the assumption 1. Simple exponential growth model 1.that Simple random1. Simple exponential (stochastic) exponential growth events growth aHistorically,ff modelect the model biological the first system, attempts in which to describe case a the model biological can only processes predict were the the population dynamics models. probability of various outcomesHistorically, inOne the future. of the the famous We first will attempts model focus is on Malthusian to deterministic describe growth the models. biological model (1798), processes sometimes were the called population a simple dynamics exponential models. growth Historically,Historically, the first the attempts first attempts to describe to describe the biological the biological processes processes were the were population the population dynamics dynamics models. models. Onemodel. of the This famous model model is essentially is Malthusian exponential growth model growth (1798), based sometimes on a constant called rate. a simple exponential growth One1. Simple ofOne the of famous exponential the famous model model is growth Malthusian is Malthusian model growth growth model model (1798), (1798), sometimes sometimes called called a simple a simple exponential exponential growth growth model. This model is essentially exponential growth based on a constant rate. model.model. This model This model is essentially is essentially exponentialLet ‘B’ exponential be, growth for example, growth based the basedon size a constant on of a a constant population rate. rate. of bacteria that can change with time. If we assume that the Historically, the first attempts to describe the biological processes were the population dynamics models. Letrate ‘B’ be, of growth for example, of the the population size of a is population proportional of bacteria to the size that of can the change population, with it time.
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