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Biological Contagion Biological Contagion Introduction Principles of Complex Systems Course 300, Fall, 2008 Simple disease spreading models Background Prediction References Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/58 Outline Biological Contagion Introduction Simple disease spreading models Background Introduction Prediction References Simple disease spreading models Background Prediction References Frame 2/58 Contagion Biological Contagion Introduction Simple disease spreading models Background Prediction A confusion of contagions: I Is Harry Potter some kind of virus? I What about the Da Vinci Code? I Does Sudoku spread like a disease? I Religion? I Democracy...? References Frame 3/58 Contagion Biological Contagion Introduction Simple disease spreading models Background Prediction A confusion of contagions: I Is Harry Potter some kind of virus? I What about the Da Vinci Code? I Does Sudoku spread like a disease? I Religion? I Democracy...? References Frame 3/58 Contagion Biological Contagion Introduction Simple disease spreading models Background Prediction A confusion of contagions: I Is Harry Potter some kind of virus? I What about the Da Vinci Code? I Does Sudoku spread like a disease? I Religion? I Democracy...? References Frame 3/58 Contagion Biological Contagion Introduction Simple disease spreading models Background Prediction A confusion of contagions: I Is Harry Potter some kind of virus? I What about the Da Vinci Code? I Does Sudoku spread like a disease? I Religion? I Democracy...? References Frame 3/58 Contagion Biological Contagion Introduction Simple disease spreading models Background Prediction A confusion of contagions: I Is Harry Potter some kind of virus? I What about the Da Vinci Code? I Does Sudoku spread like a disease? I Religion? I Democracy...? References Frame 3/58 Contagion Biological Contagion Introduction Simple disease spreading models Background Prediction A confusion of contagions: I Is Harry Potter some kind of virus? I What about the Da Vinci Code? I Does Sudoku spread like a disease? I Religion? I Democracy...? References Frame 3/58 Contagion Biological Contagion Introduction Simple disease spreading models Naturomorphisms Background Prediction References I “The feeling was contagious.” I “The news spread like wildfire.” I “Freedom is the most contagious virus known to man.” —Hubert H. Humphrey, Johnson’s vice president I “Nothing is so contagious as enthusiasm.” —Samuel Taylor Coleridge Frame 4/58 Contagion Biological Contagion Introduction Simple disease spreading models Naturomorphisms Background Prediction References I “The feeling was contagious.” I “The news spread like wildfire.” I “Freedom is the most contagious virus known to man.” —Hubert H. Humphrey, Johnson’s vice president I “Nothing is so contagious as enthusiasm.” —Samuel Taylor Coleridge Frame 4/58 Contagion Biological Contagion Introduction Simple disease spreading models Naturomorphisms Background Prediction References I “The feeling was contagious.” I “The news spread like wildfire.” I “Freedom is the most contagious virus known to man.” —Hubert H. Humphrey, Johnson’s vice president I “Nothing is so contagious as enthusiasm.” —Samuel Taylor Coleridge Frame 4/58 Contagion Biological Contagion Introduction Simple disease spreading models Naturomorphisms Background Prediction References I “The feeling was contagious.” I “The news spread like wildfire.” I “Freedom is the most contagious virus known to man.” —Hubert H. Humphrey, Johnson’s vice president I “Nothing is so contagious as enthusiasm.” —Samuel Taylor Coleridge Frame 4/58 Contagion Biological Contagion Introduction Simple disease spreading models Naturomorphisms Background Prediction References I “The feeling was contagious.” I “The news spread like wildfire.” I “Freedom is the most contagious virus known to man.” —Hubert H. Humphrey, Johnson’s vice president I “Nothing is so contagious as enthusiasm.” —Samuel Taylor Coleridge Frame 4/58 Social contagion Biological Contagion Introduction Simple disease spreading models Background Prediction References Optimism according to Ambrose Bierce: () The doctrine that everything is beautiful, including what is ugly, everything good, especially the bad, and everything right that is wrong. ... Frame 5/58 Social contagion Biological Contagion Introduction Simple disease spreading models Background Prediction References Optimism according to Ambrose Bierce: () The doctrine that everything is beautiful, including what is ugly, everything good, especially the bad, and everything right that is wrong. ... It is hereditary, but fortunately not contagious. Frame 5/58 Social contagion Biological Contagion Introduction Eric Hoffer, 1902–1983 There is a grandeur in the uniformity of the mass. When a fashion, a dance, a song, a slogan or a joke sweeps like wildfire from one end of the continent to the other, and a hundred million people roar with laughter, sway their bodies in unison, hum one song or break forth in anger and denunciation, there is the overpowering feeling that in this country we have come nearer the brotherhood of man than ever before. I Simple disease spreading models Background Prediction References Hoffer () was an interesting fellow... Frame 6/58 Social contagion Biological Contagion Introduction Eric Hoffer, 1902–1983 There is a grandeur in the uniformity of the mass. When a fashion, a dance, a song, a slogan or a joke sweeps like wildfire from one end of the continent to the other, and a hundred million people roar with laughter, sway their bodies in unison, hum one song or break forth in anger and denunciation, there is the overpowering feeling that in this country we have come nearer the brotherhood of man than ever before. I Simple disease spreading models Background Prediction References Hoffer () was an interesting fellow... Frame 6/58 The spread of fanaticism Biological Contagion Introduction Hoffer’s acclaimed work: “The True Believer: Thoughts On The Nature Of Mass Movements” (1951) [3] Simple disease spreading models Background Prediction References Quotes-aplenty: I “We can be absolutely certain only about things we do not understand.” I “Mass movements can rise and spread without belief in a God, but never without belief in a devil.” I “Where freedom is real, equality is the passion of the masses. Where equality is real, freedom is the passion of a small minority.” Frame 7/58 The spread of fanaticism Biological Contagion Introduction Hoffer’s acclaimed work: “The True Believer: Thoughts On The Nature Of Mass Movements” (1951) [3] Simple disease spreading models Background Prediction References Quotes-aplenty: I “We can be absolutely certain only about things we do not understand.” I “Mass movements can rise and spread without belief in a God, but never without belief in a devil.” I “Where freedom is real, equality is the passion of the masses. Where equality is real, freedom is the passion of a small minority.” Frame 7/58 The spread of fanaticism Biological Contagion Introduction Hoffer’s acclaimed work: “The True Believer: Thoughts On The Nature Of Mass Movements” (1951) [3] Simple disease spreading models Background Prediction References Quotes-aplenty: I “We can be absolutely certain only about things we do not understand.” I “Mass movements can rise and spread without belief in a God, but never without belief in a devil.” I “Where freedom is real, equality is the passion of the masses. Where equality is real, freedom is the passion of a small minority.” Frame 7/58 The spread of fanaticism Biological Contagion Introduction Hoffer’s acclaimed work: “The True Believer: Thoughts On The Nature Of Mass Movements” (1951) [3] Simple disease spreading models Background Prediction References Quotes-aplenty: I “We can be absolutely certain only about things we do not understand.” I “Mass movements can rise and spread without belief in a God, but never without belief in a devil.” I “Where freedom is real, equality is the passion of the masses. Where equality is real, freedom is the passion of a small minority.” Frame 7/58 The spread of fanaticism Biological Contagion Introduction Hoffer’s acclaimed work: “The True Believer: Thoughts On The Nature Of Mass Movements” (1951) [3] Simple disease spreading models Background Prediction References Quotes-aplenty: I “We can be absolutely certain only about things we do not understand.” I “Mass movements can rise and spread without belief in a God, but never without belief in a devil.” I “Where freedom is real, equality is the passion of the masses. Where equality is real, freedom is the passion of a small minority.” Frame 7/58 The spread of fanaticism Biological Contagion Introduction Hoffer’s acclaimed work: “The True Believer: Thoughts On The Nature Of Mass Movements” (1951) [3] Simple disease spreading models Background Prediction References Quotes-aplenty: I “We can be absolutely certain only about things we do not understand.” I “Mass movements can rise and spread without belief in a God, but never without belief in a devil.” I “Where freedom is real, equality is the passion of the masses. Where equality is real, freedom is the passion of a small minority.” Frame 7/58 Biological Contagion Imitation Introduction Simple disease spreading models Background Prediction “When people are free to do as they please, they usually imitate each other.” References —Eric Hoffer “The Passionate State of Mind” [4] despair.com Frame 8/58 Biological Contagion The collective... Introduction Simple disease spreading models Background Prediction References “Never Underestimate the Power of Stupid People in Large Groups.” despair.com Frame 9/58 Contagion Biological Contagion Introduction Definitions I (1) The spreading of a quality or quantity between individuals in a population. I (2) A disease itself: the plague, a blight, the dreaded lurgi, ... I from Latin: con = ‘together with’ + tangere ‘to touch.’ I Contagion has unpleasant overtones... I Just Spreading might be a more neutral word I But contagion is kind of exciting... Simple disease spreading models Background Prediction References Frame 10/58 Contagion Biological Contagion Introduction Definitions I (1) The spreading of a quality or quantity between individuals in a population. I (2) A disease itself: the plague, a blight, the dreaded lurgi, ... I from Latin: con = ‘together with’ + tangere ‘to touch.’ I Contagion has unpleasant overtones... I Just Spreading might be a more neutral word I But contagion is kind of exciting... Simple disease spreading models Background Prediction References Frame 10/58 Contagion Biological Contagion Introduction Definitions I (1) The spreading of a quality or quantity between individuals in a population. I (2) A disease itself: the plague, a blight, the dreaded lurgi, ... I from Latin: con = ‘together with’ + tangere ‘to touch.’ I Contagion has unpleasant overtones... I Just Spreading might be a more neutral word I But contagion is kind of exciting... Simple disease spreading models Background Prediction References Frame 10/58 Contagion Biological Contagion Introduction Definitions I (1) The spreading of a quality or quantity between individuals in a population. I (2) A disease itself: the plague, a blight, the dreaded lurgi, ... I from Latin: con = ‘together with’ + tangere ‘to touch.’ I Contagion has unpleasant overtones... I Just Spreading might be a more neutral word I But contagion is kind of exciting... Simple disease spreading models Background Prediction References Frame 10/58 Contagion Biological Contagion Introduction Definitions I (1) The spreading of a quality or quantity between individuals in a population. I (2) A disease itself: the plague, a blight, the dreaded lurgi, ... I from Latin: con = ‘together with’ + tangere ‘to touch.’ I Contagion has unpleasant overtones... I Just Spreading might be a more neutral word I But contagion is kind of exciting... Simple disease spreading models Background Prediction References Frame 10/58 Contagion Biological Contagion Introduction Definitions I (1) The spreading of a quality or quantity between individuals in a population. I (2) A disease itself: the plague, a blight, the dreaded lurgi, ... I from Latin: con = ‘together with’ + tangere ‘to touch.’ I Contagion has unpleasant overtones... I Just Spreading might be a more neutral word I But contagion is kind of exciting... Simple disease spreading models Background Prediction References Frame 10/58 Contagion Biological Contagion Introduction Definitions I (1) The spreading of a quality or quantity between individuals in a population. I (2) A disease itself: the plague, a blight, the dreaded lurgi, ... I from Latin: con = ‘together with’ + tangere ‘to touch.’ I Contagion has unpleasant overtones... I Just Spreading might be a more neutral word I But contagion is kind of exciting... Simple disease spreading models Background Prediction References Frame 10/58 Examples of non-disease spreading: Biological Contagion Introduction Simple disease spreading models Background Prediction References Interesting infections: I Spreading of buildings in the US. () I Spreading of spreading (). I Viral get-out-the-vote video. () Frame 11/58 Examples of non-disease spreading: Biological Contagion Introduction Simple disease spreading models Background Prediction References Interesting infections: I Spreading of buildings in the US. () I Spreading of spreading (). I Viral get-out-the-vote video. () Frame 11/58 Examples of non-disease spreading: Biological Contagion Introduction Simple disease spreading models Background Prediction References Interesting infections: I Spreading of buildings in the US. () I Spreading of spreading (). I Viral get-out-the-vote video. () Frame 11/58 Examples of non-disease spreading: Biological Contagion Introduction Simple disease spreading models Background Prediction References Interesting infections: I Spreading of buildings in the US. () I Spreading of spreading (). I Viral get-out-the-vote video. () Frame 11/58 Contagions Biological Contagion Introduction Simple disease spreading models Background Prediction Two main classes of contagion References 1. Infectious diseases 2. Social contagion Frame 12/58 Contagions Biological Contagion Introduction Simple disease spreading models Background Prediction Two main classes of contagion References 1. Infectious diseases 2. Social contagion Frame 12/58 Contagions Biological Contagion Introduction Simple disease spreading models Background Prediction Two main classes of contagion References 1. Infectious diseases 2. Social contagion Frame 12/58 Contagions Biological Contagion Introduction Simple disease spreading models Background Prediction Two main classes of contagion References 1. Infectious diseases: tuberculosis, HIV, ebola, SARS, influenza, ... 2. Social contagion Frame 12/58 Contagions Biological Contagion Introduction Simple disease spreading models Background Prediction Two main classes of contagion References 1. Infectious diseases: tuberculosis, HIV, ebola, SARS, influenza, ... 2. Social contagion: fashion, word usage, rumors, riots, religion, ... Frame 12/58 Outline Biological Contagion Introduction Simple disease spreading models Background Introduction Prediction References Simple disease spreading models Background Prediction References Frame 13/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered or Removed or Refractory I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered or Removed or Refractory I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered or Removed or Refractory I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered or Removed or Refractory I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction The standard SIR model [8] Simple disease spreading models Background I I The basic model of disesase contagion Three states: Prediction References 1. S = Susceptible 2. I = Infective/Infectious 3. R = Recovered or Removed or Refractory I S(t) + I(t) + R(t) = 1 I Presumes random interactions (mass-action principle) I Interactions are independent (no memory) I Discrete and continuous time versions Frame 14/58 Mathematical Epidemiology Biological Contagion Introduction Discrete time automata example: 1 − βI Simple disease spreading models Background Prediction References S βI ρ I r R 1−r 1−ρ Frame 15/58 Mathematical Epidemiology Biological Contagion Introduction Discrete time automata example: 1 − βI Simple disease spreading models Background Prediction References S Transition Probabilities: βI ρ I r R 1−r 1−ρ Frame 15/58 Mathematical Epidemiology Biological Contagion Introduction Discrete time automata example: 1 − βI Background Prediction References S Transition Probabilities: βI ρ β for being infected given contact with infected I r R Simple disease spreading models 1−r 1−ρ Frame 15/58 Mathematical Epidemiology Biological Contagion Introduction Discrete time automata example: 1 − βI Background Prediction References S Transition Probabilities: βI ρ β for being infected given contact with infected r for recovery I r R Simple disease spreading models 1−r 1−ρ Frame 15/58 Mathematical Epidemiology Biological Contagion Introduction Discrete time automata example: 1 − βI Background Prediction References S Transition Probabilities: βI ρ I r R Simple disease spreading models 1−r β for being infected given contact with infected r for recovery ρ for loss of immunity 1−ρ Frame 15/58 Mathematical Epidemiology Biological Contagion Introduction Simple disease spreading models Background Prediction Original models attributed to I 1920’s: Reed and Frost I 1920’s/1930’s: Kermack and McKendrick [5, 7, 6] I Coupled differential equations with a mass-action principle References Frame 16/58 Mathematical Epidemiology Biological Contagion Introduction Simple disease spreading models Background Prediction Original models attributed to I 1920’s: Reed and Frost I 1920’s/1930’s: Kermack and McKendrick [5, 7, 6] I Coupled differential equations with a mass-action principle References Frame 16/58 Mathematical Epidemiology Biological Contagion Introduction Simple disease spreading models Background Prediction Original models attributed to I 1920’s: Reed and Frost I 1920’s/1930’s: Kermack and McKendrick [5, 7, 6] I Coupled differential equations with a mass-action principle References Frame 16/58 Mathematical Epidemiology Biological Contagion Introduction Simple disease spreading models Background Prediction Original models attributed to I 1920’s: Reed and Frost I 1920’s/1930’s: Kermack and McKendrick [5, 7, 6] I Coupled differential equations with a mass-action principle References Frame 16/58 Independent Interaction models Differential equations for continuous model d S = −βIS + ρR dt d I = βIS − rI dt d R = rI − ρR dt Biological Contagion Introduction Simple disease spreading models Background Prediction References β, r , and ρ are now rates. Reproduction Number R0 : I R0 = expected number of infected individuals resulting from a single initial infective I Epidemic threshold: If R0 > 1, ‘epidemic’ occurs. Frame 17/58 Independent Interaction models Differential equations for continuous model d S = −βIS + ρR dt d I = βIS − rI dt d R = rI − ρR dt Biological Contagion Introduction Simple disease spreading models Background Prediction References β, r , and ρ are now rates. Reproduction Number R0 : I R0 = expected number of infected individuals resulting from a single initial infective I Epidemic threshold: If R0 > 1, ‘epidemic’ occurs. Frame 17/58 Independent Interaction models Differential equations for continuous model d S = −βIS + ρR dt d I = βIS − rI dt d R = rI − ρR dt Biological Contagion Introduction Simple disease spreading models Background Prediction References β, r , and ρ are now rates. Reproduction Number R0 : I R0 = expected number of infected individuals resulting from a single initial infective I Epidemic threshold: If R0 > 1, ‘epidemic’ occurs. Frame 17/58 Independent Interaction models Differential equations for continuous model d S = −βIS + ρR dt d I = βIS − rI dt d R = rI − ρR dt Biological Contagion Introduction Simple disease spreading models Background Prediction References β, r , and ρ are now rates. Reproduction Number R0 : I R0 = expected number of infected individuals resulting from a single initial infective I Epidemic threshold: If R0 > 1, ‘epidemic’ occurs. Frame 17/58 Reproduction Number R0 Biological Contagion Introduction Discrete version: Simple disease spreading models Background Prediction I Set up: One Infective in a randomly mixing population of Susceptibles I At time t = 0, single infective random bumps into a Susceptible I Probability of transmission = β I At time t = 1, single Infective remains infected with probability 1 − r I At time t = k , single Infective remains infected with probability (1 − r )k References Frame 18/58 Reproduction Number R0 Biological Contagion Introduction Discrete version: Simple disease spreading models Background Prediction I Set up: One Infective in a randomly mixing population of Susceptibles I At time t = 0, single infective random bumps into a Susceptible I Probability of transmission = β I At time t = 1, single Infective remains infected with probability 1 − r I At time t = k , single Infective remains infected with probability (1 − r )k References Frame 18/58 Reproduction Number R0 Biological Contagion Introduction Discrete version: Simple disease spreading models Background Prediction I Set up: One Infective in a randomly mixing population of Susceptibles I At time t = 0, single infective random bumps into a Susceptible I Probability of transmission = β I At time t = 1, single Infective remains infected with probability 1 − r I At time t = k , single Infective remains infected with probability (1 − r )k References Frame 18/58 Reproduction Number R0 Biological Contagion Introduction Discrete version: Simple disease spreading models Background Prediction I Set up: One Infective in a randomly mixing population of Susceptibles I At time t = 0, single infective random bumps into a Susceptible I Probability of transmission = β I At time t = 1, single Infective remains infected with probability 1 − r I At time t = k , single Infective remains infected with probability (1 − r )k References Frame 18/58 Reproduction Number R0 Biological Contagion Introduction Discrete version: Simple disease spreading models Background Prediction I Set up: One Infective in a randomly mixing population of Susceptibles I At time t = 0, single infective random bumps into a Susceptible I Probability of transmission = β I At time t = 1, single Infective remains infected with probability 1 − r I At time t = k , single Infective remains infected with probability (1 − r )k References Frame 18/58 Biological Contagion Reproduction Number R0 Discrete version: I Introduction Expected number infected by original Infective: Simple disease spreading models Background Prediction 2 3 R0 = β + (1 − r )β + (1 − r ) β + (1 − r ) β + . . . References Frame 19/58 Biological Contagion Reproduction Number R0 Discrete version: I Introduction Expected number infected by original Infective: Simple disease spreading models Background Prediction 2 3 R0 = β + (1 − r )β + (1 − r ) β + (1 − r ) β + . . . References = β 1 + (1 − r ) + (1 − r )2 + (1 − r )3 + . . . Frame 19/58 Biological Contagion Reproduction Number R0 Discrete version: I Introduction Expected number infected by original Infective: Simple disease spreading models Background Prediction 2 3 R0 = β + (1 − r )β + (1 − r ) β + (1 − r ) β + . . . References = β 1 + (1 − r ) + (1 − r )2 + (1 − r )3 + . . . =β 1 1 − (1 − r ) Frame 19/58 Biological Contagion Reproduction Number R0 Discrete version: I Introduction Expected number infected by original Infective: Simple disease spreading models Background Prediction 2 3 R0 = β + (1 − r )β + (1 − r ) β + (1 − r ) β + . . . References = β 1 + (1 − r ) + (1 − r )2 + (1 − r )3 + . . . =β 1 = β/r 1 − (1 − r ) Frame 19/58 Biological Contagion Reproduction Number R0 Discrete version: I Introduction Expected number infected by original Infective: Simple disease spreading models Background Prediction 2 3 R0 = β + (1 − r )β + (1 − r ) β + (1 − r ) β + . . . References = β 1 + (1 − r ) + (1 − r )2 + (1 − r )3 + . . . =β 1 = β/r 1 − (1 − r ) For S0 initial infectives (1 − S0 = R0 immune): R0 = S0 β/r Frame 19/58 Independent Interaction models For the continuous version I Biological Contagion Introduction Simple disease spreading models Second equation: Background Prediction d I = βSI − rI dt I References Number of infectives grows initially if βS(0) − r > 0 I Same story as for discrete model. Frame 20/58 Independent Interaction models For the continuous version I Biological Contagion Introduction Simple disease spreading models Second equation: Background Prediction d I = βSI − rI dt References d I = (βS − r )I dt I Number of infectives grows initially if βS(0) − r > 0 I Same story as for discrete model. Frame 20/58 Independent Interaction models For the continuous version I Biological Contagion Introduction Simple disease spreading models Second equation: Background Prediction d I = βSI − rI dt References d I = (βS − r )I dt I Number of infectives grows initially if βS(0) − r > 0 I Same story as for discrete model. Frame 20/58 Independent Interaction models For the continuous version I Biological Contagion Introduction Simple disease spreading models Second equation: Background Prediction d I = βSI − rI dt References d I = (βS − r )I dt I Number of infectives grows initially if βS(0) − r > 0 ⇒ βS(0) > r I Same story as for discrete model. Frame 20/58 Independent Interaction models For the continuous version I Biological Contagion Introduction Simple disease spreading models Second equation: Background Prediction d I = βSI − rI dt References d I = (βS − r )I dt I Number of infectives grows initially if βS(0) − r > 0 ⇒ βS(0) > r ⇒ βS(0)/r > 1 I Same story as for discrete model. Frame 20/58 Independent Interaction models For the continuous version I Biological Contagion Introduction Simple disease spreading models Second equation: Background Prediction d I = βSI − rI dt References d I = (βS − r )I dt I Number of infectives grows initially if βS(0) − r > 0 ⇒ βS(0) > r ⇒ βS(0)/r > 1 I Same story as for discrete model. Frame 20/58 Biological Contagion Independent Interaction models Introduction Example of epidemic threshold: Simple disease spreading models 1 Background Fraction infected Prediction 0.8 References 0.6 0.4 0.2 0 0 1 2 3 4 R0 Frame 21/58 Biological Contagion Independent Interaction models Introduction Example of epidemic threshold: Simple disease spreading models 1 Background Fraction infected Prediction 0.8 References 0.6 0.4 0.2 0 0 1 2 3 4 R0 I Continuous phase transition. Frame 21/58 Biological Contagion Independent Interaction models Introduction Example of epidemic threshold: Simple disease spreading models 1 Background Fraction infected Prediction 0.8 References 0.6 0.4 0.2 0 0 1 2 3 4 R0 I Continuous phase transition. I Fine idea from a simple model. Frame 21/58 Independent Interaction models Biological Contagion Introduction Simple disease spreading models Background Prediction Many variants of the SIR model: I SIS: susceptible-infective-susceptible I SIRS: susceptible-infective-recovered-susceptible I compartment models (age or gender partitions) I more categories such as ‘exposed’ (SEIRS) I recruitment (migration, birth) References Frame 22/58 Independent Interaction models Biological Contagion Introduction Simple disease spreading models Background Prediction Many variants of the SIR model: I SIS: susceptible-infective-susceptible I SIRS: susceptible-infective-recovered-susceptible I compartment models (age or gender partitions) I more categories such as ‘exposed’ (SEIRS) I recruitment (migration, birth) References Frame 22/58 Independent Interaction models Biological Contagion Introduction Simple disease spreading models Background Prediction Many variants of the SIR model: I SIS: susceptible-infective-susceptible I SIRS: susceptible-infective-recovered-susceptible I compartment models (age or gender partitions) I more categories such as ‘exposed’ (SEIRS) I recruitment (migration, birth) References Frame 22/58 Independent Interaction models Biological Contagion Introduction Simple disease spreading models Background Prediction Many variants of the SIR model: I SIS: susceptible-infective-susceptible I SIRS: susceptible-infective-recovered-susceptible I compartment models (age or gender partitions) I more categories such as ‘exposed’ (SEIRS) I recruitment (migration, birth) References Frame 22/58 Independent Interaction models Biological Contagion Introduction Simple disease spreading models Background Prediction Many variants of the SIR model: I SIS: susceptible-infective-susceptible I SIRS: susceptible-infective-recovered-susceptible I compartment models (age or gender partitions) I more categories such as ‘exposed’ (SEIRS) I recruitment (migration, birth) References Frame 22/58 Independent Interaction models Biological Contagion Introduction Simple disease spreading models Background Prediction Many variants of the SIR model: I SIS: susceptible-infective-susceptible I SIRS: susceptible-infective-recovered-susceptible I compartment models (age or gender partitions) I more categories such as ‘exposed’ (SEIRS) I recruitment (migration, birth) References Frame 22/58 Outline Biological Contagion Introduction Simple disease spreading models Background Introduction Prediction References Simple disease spreading models Background Prediction References Frame 23/58 Disease spreading models Biological Contagion Introduction Simple disease spreading models Background Prediction References For novel diseases: 1. Can we predict the size of an epidemic? 2. How important is the reproduction number R0 ? Frame 24/58 Disease spreading models Biological Contagion Introduction Simple disease spreading models Background Prediction References For novel diseases: 1. Can we predict the size of an epidemic? 2. How important is the reproduction number R0 ? Frame 24/58 Disease spreading models Biological Contagion Introduction Simple disease spreading models Background Prediction References For novel diseases: 1. Can we predict the size of an epidemic? 2. How important is the reproduction number R0 ? Frame 24/58 R0 and variation in epidemic sizes Biological Contagion Introduction Simple disease spreading models Background Prediction R0 approximately same for all of the following: I 1918-19 “Spanish Flu” ∼ 500,000 deaths in US I 1957-58 “Asian Flu” ∼ 70,000 deaths in US I 1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US I 2003 “SARS Epidemic” ∼ 800 deaths world-wide References Frame 25/58 R0 and variation in epidemic sizes Biological Contagion Introduction Simple disease spreading models Background Prediction R0 approximately same for all of the following: I 1918-19 “Spanish Flu” ∼ 500,000 deaths in US I 1957-58 “Asian Flu” ∼ 70,000 deaths in US I 1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US I 2003 “SARS Epidemic” ∼ 800 deaths world-wide References Frame 25/58 R0 and variation in epidemic sizes Biological Contagion Introduction Simple disease spreading models Background Prediction R0 approximately same for all of the following: I 1918-19 “Spanish Flu” ∼ 500,000 deaths in US I 1957-58 “Asian Flu” ∼ 70,000 deaths in US I 1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US I 2003 “SARS Epidemic” ∼ 800 deaths world-wide References Frame 25/58 R0 and variation in epidemic sizes Biological Contagion Introduction Simple disease spreading models Background Prediction R0 approximately same for all of the following: I 1918-19 “Spanish Flu” ∼ 500,000 deaths in US I 1957-58 “Asian Flu” ∼ 70,000 deaths in US I 1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US I 2003 “SARS Epidemic” ∼ 800 deaths world-wide References Frame 25/58 R0 and variation in epidemic sizes Biological Contagion Introduction Simple disease spreading models Background Prediction R0 approximately same for all of the following: I 1918-19 “Spanish Flu” ∼ 500,000 deaths in US I 1957-58 “Asian Flu” ∼ 70,000 deaths in US I 1968-69 “Hong Kong Flu” ∼ 34,000 deaths in US I 2003 “SARS Epidemic” ∼ 800 deaths world-wide References Frame 25/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Size distributions are important elsewhere: I earthquakes (Gutenberg-Richter law) I city sizes, forest fires, war fatalities I wealth distributions I ‘popularity’ (books, music, websites, ideas) I Epidemics? Prediction References Frame 26/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Size distributions are important elsewhere: I earthquakes (Gutenberg-Richter law) I city sizes, forest fires, war fatalities I wealth distributions I ‘popularity’ (books, music, websites, ideas) I Epidemics? Prediction References Frame 26/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Size distributions are important elsewhere: I earthquakes (Gutenberg-Richter law) I city sizes, forest fires, war fatalities I wealth distributions I ‘popularity’ (books, music, websites, ideas) I Epidemics? Prediction References Frame 26/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Size distributions are important elsewhere: I earthquakes (Gutenberg-Richter law) I city sizes, forest fires, war fatalities I wealth distributions I ‘popularity’ (books, music, websites, ideas) I Epidemics? Prediction References Power laws distributions are common but not obligatory... Frame 26/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Prediction References Really, what about epidemics? I Simply hasn’t attracted much attention. I Data not as clean as for other phenomena. Frame 27/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Prediction References Really, what about epidemics? I Simply hasn’t attracted much attention. I Data not as clean as for other phenomena. Frame 27/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Prediction References Really, what about epidemics? I Simply hasn’t attracted much attention. I Data not as clean as for other phenomena. Frame 27/58 Feeling Ill in Iceland Biological Contagion Introduction Simple disease spreading models Caseload recorded monthly for range of diseases in Iceland, 1888-1890 Background Prediction References Frequency 0.03 0.02 Iceland: measles normalized count 0.01 0 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 Date Treat outbreaks separated in time as ‘novel’ diseases. Frame 28/58 Biological Contagion Really not so good at all in Iceland Introduction Simple disease spreading models Background Epidemic size distributions N(S) for Measles, Rubella, and Whooping Cough. 75 105 N(S) B C 5 5 5 4 4 4 3 3 3 2 2 2 1 1 0.025 0.05 S 0.075 References 75 A 0 0 Prediction 0.1 0 0 1 0.02 0.04 0.06 0 0 S 0.025 0.05 0.075 S Spike near S = 0, relatively flat otherwise. Frame 29/58 Biological Contagion Measles & Pertussis 2 N (ψ) N>(ψ) 10 A 1 5 10 10 −5 10 −4 10 −3 10 3 −2 10 4 −1 10 2 1 1 0.025 B 1 10 Background 0.05 ψ 0.075 10 −5 10 −4 10 −3 10 −2 10 3 ψ 2 0 0 Simple disease spreading models 0 0 4 N>(ψ) 10 2 5 Introduction 75 75 0.1 0 0 Prediction −1 10 References ψ 0.025 0.05 0.075 ψ Frame 30/58 Biological Contagion Measles & Pertussis 2 N (ψ) N>(ψ) 10 A 1 5 10 10 −5 10 −4 10 −3 10 3 −2 10 4 −1 10 2 1 1 0.025 B 1 10 Background 0.05 0.075 10 −5 10 −4 10 −3 10 −2 10 3 ψ 2 0 0 Simple disease spreading models 0 0 4 N>(ψ) 10 2 5 Introduction 75 75 0.1 0 0 Prediction −1 10 References ψ 0.025 0.05 0.075 ψ ψ Insert plots: Complementary cumulative frequency distributions: N(Ψ0 > Ψ) ∝ Ψ−γ+1 Limited scaling with a possible break. Frame 30/58 Power law distributions Biological Contagion Introduction Simple disease spreading models Background Measured values of γ: I measles: 1.40 (low Ψ) and 1.13 (high Ψ) I pertussis: 1.39 (low Ψ) and 1.16 (high Ψ) I Expect 2 ≤ γ < 3 (finite mean, infinite variance) I When γ < 1, can’t normalize I Distribution is quite flat. Prediction References Frame 31/58 Power law distributions Biological Contagion Introduction Simple disease spreading models Background Measured values of γ: I measles: 1.40 (low Ψ) and 1.13 (high Ψ) I pertussis: 1.39 (low Ψ) and 1.16 (high Ψ) I Expect 2 ≤ γ < 3 (finite mean, infinite variance) I When γ < 1, can’t normalize I Distribution is quite flat. Prediction References Frame 31/58 Power law distributions Biological Contagion Introduction Simple disease spreading models Background Measured values of γ: I measles: 1.40 (low Ψ) and 1.13 (high Ψ) I pertussis: 1.39 (low Ψ) and 1.16 (high Ψ) I Expect 2 ≤ γ < 3 (finite mean, infinite variance) I When γ < 1, can’t normalize I Distribution is quite flat. Prediction References Frame 31/58 Power law distributions Biological Contagion Introduction Simple disease spreading models Background Measured values of γ: I measles: 1.40 (low Ψ) and 1.13 (high Ψ) I pertussis: 1.39 (low Ψ) and 1.16 (high Ψ) I Expect 2 ≤ γ < 3 (finite mean, infinite variance) I When γ < 1, can’t normalize I Distribution is quite flat. Prediction References Frame 31/58 Power law distributions Biological Contagion Introduction Simple disease spreading models Background Measured values of γ: I measles: 1.40 (low Ψ) and 1.13 (high Ψ) I pertussis: 1.39 (low Ψ) and 1.16 (high Ψ) I Expect 2 ≤ γ < 3 (finite mean, infinite variance) I When γ < 1, can’t normalize I Distribution is quite flat. Prediction References Frame 31/58 Power law distributions Biological Contagion Introduction Simple disease spreading models Background Measured values of γ: I measles: 1.40 (low Ψ) and 1.13 (high Ψ) I pertussis: 1.39 (low Ψ) and 1.16 (high Ψ) I Expect 2 ≤ γ < 3 (finite mean, infinite variance) I When γ < 1, can’t normalize I Distribution is quite flat. Prediction References Frame 31/58 Biological Contagion Resurgence—example of SARS Introduction 160 # New cases D 120 Simple disease spreading models Background Prediction 80 References 40 0 Nov 16, ’02 Dec 16, ’02 Jan 15, ’03 Feb 14, ’03 Mar 16, ’03 Apr 15, ’03 May 15, ’03 Jun 14, ’03 Date of onset I Epidemic slows... I Epidemic discovers new ‘pools’ of susceptibles: Resurgence. I Importance of rare, stochastic events. Frame 32/58 Biological Contagion Resurgence—example of SARS Introduction 160 # New cases D 120 Simple disease spreading models Background Prediction 80 References 40 0 Nov 16, ’02 Dec 16, ’02 Jan 15, ’03 Feb 14, ’03 Mar 16, ’03 Apr 15, ’03 May 15, ’03 Jun 14, ’03 Date of onset I Epidemic slows... I Epidemic discovers new ‘pools’ of susceptibles: Resurgence. I Importance of rare, stochastic events. Frame 32/58 Biological Contagion Resurgence—example of SARS Introduction 160 # New cases D 120 Simple disease spreading models Background Prediction 80 References 40 0 Nov 16, ’02 Dec 16, ’02 Jan 15, ’03 Feb 14, ’03 Mar 16, ’03 Apr 15, ’03 May 15, ’03 Jun 14, ’03 Date of onset I Epidemic slows... then an infective moves to a new context. I Epidemic discovers new ‘pools’ of susceptibles: Resurgence. I Importance of rare, stochastic events. Frame 32/58 Biological Contagion Resurgence—example of SARS Introduction 160 # New cases D 120 Simple disease spreading models Background Prediction 80 References 40 0 Nov 16, ’02 Dec 16, ’02 Jan 15, ’03 Feb 14, ’03 Mar 16, ’03 Apr 15, ’03 May 15, ’03 Jun 14, ’03 Date of onset I Epidemic slows... then an infective moves to a new context. I Epidemic discovers new ‘pools’ of susceptibles: Resurgence. I Importance of rare, stochastic events. Frame 32/58 Biological Contagion Resurgence—example of SARS Introduction 160 # New cases D 120 Simple disease spreading models Background Prediction 80 References 40 0 Nov 16, ’02 Dec 16, ’02 Jan 15, ’03 Feb 14, ’03 Mar 16, ’03 Apr 15, ’03 May 15, ’03 Jun 14, ’03 Date of onset I Epidemic slows... then an infective moves to a new context. I Epidemic discovers new ‘pools’ of susceptibles: Resurgence. I Importance of rare, stochastic events. Frame 32/58 The challenge Biological Contagion Introduction Simple disease spreading models Background Prediction References So... can a simple model produce 1. broad epidemic distributions and 2. resurgence ? Frame 33/58 Biological Contagion Size distributions Introduction 2000 A R0=3 1500 N(ψ) Simple disease spreading models Background Simple models typically produce bimodal or unimodal size distributions. 1000 500 0 0 0.25 0.5 0.75 Prediction References 1 ψ I I This includes network models: random, small-world, scale-free, ... Exceptions: 1. Forest fire models 2. Sophisticated metapopulation models Frame 34/58 Biological Contagion Size distributions Introduction 2000 A R0=3 1500 N(ψ) Simple disease spreading models Background Simple models typically produce bimodal or unimodal size distributions. 1000 500 0 0 0.25 0.5 0.75 Prediction References 1 ψ I I This includes network models: random, small-world, scale-free, ... Exceptions: 1. Forest fire models 2. Sophisticated metapopulation models Frame 34/58 Biological Contagion Size distributions Introduction 2000 A R0=3 1500 N(ψ) Simple disease spreading models Background Simple models typically produce bimodal or unimodal size distributions. 1000 500 0 0 0.25 0.5 0.75 Prediction References 1 ψ I I This includes network models: random, small-world, scale-free, ... Exceptions: 1. Forest fire models 2. Sophisticated metapopulation models Frame 34/58 Biological Contagion Size distributions Introduction 2000 A R0=3 1500 N(ψ) Simple disease spreading models Background Simple models typically produce bimodal or unimodal size distributions. 1000 500 0 0 0.25 0.5 0.75 Prediction References 1 ψ I I This includes network models: random, small-world, scale-free, ... Exceptions: 1. Forest fire models 2. Sophisticated metapopulation models Frame 34/58 Biological Contagion Size distributions Introduction 2000 A R0=3 1500 N(ψ) Simple disease spreading models Background Simple models typically produce bimodal or unimodal size distributions. 1000 500 0 0 0.25 0.5 0.75 Prediction References 1 ψ I I This includes network models: random, small-world, scale-free, ... Exceptions: 1. Forest fire models 2. Sophisticated metapopulation models Frame 34/58 Burning through the population Biological Contagion Introduction Forest fire models: [9] Simple disease spreading models Background I Rhodes & Anderson, 1996 I The physicist’s approach: “if it works for magnets, it’ll work for people...” Prediction References A bit of a stretch: 1. Epidemics ≡ forest fires spreading on 3-d and 5-d lattices. 2. Claim Iceland and Faroe Islands exhibit power law distributions for outbreaks. 3. Original forest fire model not completely understood. Frame 35/58 Burning through the population Biological Contagion Introduction Forest fire models: [9] Simple disease spreading models Background I Rhodes & Anderson, 1996 I The physicist’s approach: “if it works for magnets, it’ll work for people...” Prediction References A bit of a stretch: 1. Epidemics ≡ forest fires spreading on 3-d and 5-d lattices. 2. Claim Iceland and Faroe Islands exhibit power law distributions for outbreaks. 3. Original forest fire model not completely understood. Frame 35/58 Burning through the population Biological Contagion Introduction Forest fire models: [9] Simple disease spreading models Background I Rhodes & Anderson, 1996 I The physicist’s approach: “if it works for magnets, it’ll work for people...” Prediction References A bit of a stretch: 1. Epidemics ≡ forest fires spreading on 3-d and 5-d lattices. 2. Claim Iceland and Faroe Islands exhibit power law distributions for outbreaks. 3. Original forest fire model not completely understood. Frame 35/58 Burning through the population Biological Contagion Introduction Forest fire models: [9] Simple disease spreading models Background I Rhodes & Anderson, 1996 I The physicist’s approach: “if it works for magnets, it’ll work for people...” Prediction References A bit of a stretch: 1. Epidemics ≡ forest fires spreading on 3-d and 5-d lattices. 2. Claim Iceland and Faroe Islands exhibit power law distributions for outbreaks. 3. Original forest fire model not completely understood. Frame 35/58 Burning through the population Biological Contagion Introduction Forest fire models: [9] Simple disease spreading models Background I Rhodes & Anderson, 1996 I The physicist’s approach: “if it works for magnets, it’ll work for people...” Prediction References A bit of a stretch: 1. Epidemics ≡ forest fires spreading on 3-d and 5-d lattices. 2. Claim Iceland and Faroe Islands exhibit power law distributions for outbreaks. 3. Original forest fire model not completely understood. Frame 35/58 Burning through the population Biological Contagion Introduction Forest fire models: [9] Simple disease spreading models Background I Rhodes & Anderson, 1996 I The physicist’s approach: “if it works for magnets, it’ll work for people...” Prediction References A bit of a stretch: 1. Epidemics ≡ forest fires spreading on 3-d and 5-d lattices. 2. Claim Iceland and Faroe Islands exhibit power law distributions for outbreaks. 3. Original forest fire model not completely understood. Frame 35/58 Burning through the population Biological Contagion Introduction Forest fire models: [9] Simple disease spreading models Background I Rhodes & Anderson, 1996 I The physicist’s approach: “if it works for magnets, it’ll work for people...” Prediction References A bit of a stretch: 1. Epidemics ≡ forest fires spreading on 3-d and 5-d lattices. 2. Claim Iceland and Faroe Islands exhibit power law distributions for outbreaks. 3. Original forest fire model not completely understood. Frame 35/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Prediction References From Rhodes and Anderson, 1996. Frame 36/58 Sophisticated metapopulation models Biological Contagion Introduction Simple disease spreading models I Community based mixing: Longini (two scales). I Eubank et al.’s EpiSims/TRANSIMS—city simulations. I Spreading through countries—Airlines: Germann et al., Corlizza et al. I Vital work but perhaps hard to generalize from... I ⇒ Create a simple model involving multiscale travel I Multiscale models suggested by others but not formalized (Bailey, Cliff and Haggett, Ferguson et al.) Background Prediction References Frame 37/58 Sophisticated metapopulation models Biological Contagion Introduction Simple disease spreading models I Community based mixing: Longini (two scales). I Eubank et al.’s EpiSims/TRANSIMS—city simulations. I Spreading through countries—Airlines: Germann et al., Corlizza et al. I Vital work but perhaps hard to generalize from... I ⇒ Create a simple model involving multiscale travel I Multiscale models suggested by others but not formalized (Bailey, Cliff and Haggett, Ferguson et al.) Background Prediction References Frame 37/58 Sophisticated metapopulation models Biological Contagion Introduction Simple disease spreading models I Community based mixing: Longini (two scales). I Eubank et al.’s EpiSims/TRANSIMS—city simulations. I Spreading through countries—Airlines: Germann et al., Corlizza et al. I Vital work but perhaps hard to generalize from... I ⇒ Create a simple model involving multiscale travel I Multiscale models suggested by others but not formalized (Bailey, Cliff and Haggett, Ferguson et al.) Background Prediction References Frame 37/58 Sophisticated metapopulation models Biological Contagion Introduction Simple disease spreading models I Community based mixing: Longini (two scales). I Eubank et al.’s EpiSims/TRANSIMS—city simulations. I Spreading through countries—Airlines: Germann et al., Corlizza et al. I Vital work but perhaps hard to generalize from... I ⇒ Create a simple model involving multiscale travel I Multiscale models suggested by others but not formalized (Bailey, Cliff and Haggett, Ferguson et al.) Background Prediction References Frame 37/58 Sophisticated metapopulation models Biological Contagion Introduction Simple disease spreading models I Community based mixing: Longini (two scales). I Eubank et al.’s EpiSims/TRANSIMS—city simulations. I Spreading through countries—Airlines: Germann et al., Corlizza et al. I Vital work but perhaps hard to generalize from... I ⇒ Create a simple model involving multiscale travel I Multiscale models suggested by others but not formalized (Bailey, Cliff and Haggett, Ferguson et al.) Background Prediction References Frame 37/58 Sophisticated metapopulation models Biological Contagion Introduction Simple disease spreading models I Community based mixing: Longini (two scales). I Eubank et al.’s EpiSims/TRANSIMS—city simulations. I Spreading through countries—Airlines: Germann et al., Corlizza et al. I Vital work but perhaps hard to generalize from... I ⇒ Create a simple model involving multiscale travel I Multiscale models suggested by others but not formalized (Bailey, Cliff and Haggett, Ferguson et al.) Background Prediction References Frame 37/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Prediction References I Very big question: What is N? I Should we model SARS in Hong Kong as spreading I For simple models, we need to know the final size beforehand... Frame 38/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Prediction References I Very big question: What is N? I Should we model SARS in Hong Kong as spreading I For simple models, we need to know the final size beforehand... Frame 38/58 Size distributions Biological Contagion Introduction Simple disease spreading models Background Prediction References I Very big question: What is N? I Should we model SARS in Hong Kong as spreading I For simple models, we need to know the final size beforehand... Frame 38/58 Biological Contagion Improving simple models Contexts and Identities—Bipartite networks 1 2 3 contexts 4 Introduction Simple disease spreading models Background Prediction References individuals a b c b d e d unipartite network a c e I boards of directors I movies I transportation modes (subway) Frame 39/58 Biological Contagion Improving simple models Contexts and Identities—Bipartite networks 1 2 3 contexts 4 Introduction Simple disease spreading models Background Prediction References individuals a b c b d e d unipartite network a c e I boards of directors I movies I transportation modes (subway) Frame 39/58 Biological Contagion Improving simple models Contexts and Identities—Bipartite networks 1 2 3 contexts 4 Introduction Simple disease spreading models Background Prediction References individuals a b c b d e d unipartite network a c e I boards of directors I movies I transportation modes (subway) Frame 39/58 Biological Contagion Improving simple models Contexts and Identities—Bipartite networks 1 2 3 contexts 4 Introduction Simple disease spreading models Background Prediction References individuals a b c b d e d unipartite network a c e I boards of directors I movies I transportation modes (subway) Frame 39/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Improving simple models Biological Contagion Introduction Idea for social networks: incorporate identity. Identity is formed from attributes such as: I Geographic location I Type of employment I Age I Recreational activities Simple disease spreading models Background Prediction References Groups are crucial... I formed by people with at least one similar attribute I Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. [11] Frame 40/58 Biological Contagion Infer interactions/network from identities Introduction occupation Simple disease spreading models Background Prediction education high school teacher a b health care References kindergarten teacher nurse doctor c d e Distance makes sense in identity/context space. Frame 41/58 Biological Contagion Generalized context space Introduction Simple disease spreading models geography occupation age Background Prediction 0 a b c 100 d References e (Blau & Schwartz [1] , Simmel [10] , Breiger [2] ) Frame 42/58 A toy agent-based model Biological Contagion Introduction Geography—allow people to move between contexts: I Locally: standard SIR model with random mixing I discrete time simulation I β = infection probability I γ = recovery probability I P = probability of travel I Movement distance: Pr(d) ∝ exp(−d/ξ) I ξ = typical travel distance Simple disease spreading models Background Prediction References Frame 43/58 A toy agent-based model Biological Contagion Introduction Geography—allow people to move between contexts: I Locally: standard SIR model with random mixing I discrete time simulation I β = infection probability I γ = recovery probability I P = probability of travel I Movement distance: Pr(d) ∝ exp(−d/ξ) I ξ = typical travel distance Simple disease spreading models Background Prediction References Frame 43/58 A toy agent-based model Biological Contagion Introduction Geography—allow people to move between contexts: I Locally: standard SIR model with random mixing I discrete time simulation I β = infection probability I γ = recovery probability I P = probability of travel I Movement distance: Pr(d) ∝ exp(−d/ξ) I ξ = typical travel distance Simple disease spreading models Background Prediction References Frame 43/58 A toy agent-based model Biological Contagion Introduction Geography—allow people to move between contexts: I Locally: standard SIR model with random mixing I discrete time simulation I β = infection probability I γ = recovery probability I P = probability of travel I Movement distance: Pr(d) ∝ exp(−d/ξ) I ξ = typical travel distance Simple disease spreading models Background Prediction References Frame 43/58 A toy agent-based model Biological Contagion Introduction Geography—allow people to move between contexts: I Locally: standard SIR model with random mixing I discrete time simulation I β = infection probability I γ = recovery probability I P = probability of travel I Movement distance: Pr(d) ∝ exp(−d/ξ) I ξ = typical travel distance Simple disease spreading models Background Prediction References Frame 43/58 A toy agent-based model Biological Contagion Introduction Geography—allow people to move between contexts: I Locally: standard SIR model with random mixing I discrete time simulation I β = infection probability I γ = recovery probability I P = probability of travel I Movement distance: Pr(d) ∝ exp(−d/ξ) I ξ = typical travel distance Simple disease spreading models Background Prediction References Frame 43/58 A toy agent-based model Biological Contagion Introduction Geography—allow people to move between contexts: I Locally: standard SIR model with random mixing I discrete time simulation I β = infection probability I γ = recovery probability I P = probability of travel I Movement distance: Pr(d) ∝ exp(−d/ξ) I ξ = typical travel distance Simple disease spreading models Background Prediction References Frame 43/58 A toy agent-based model Biological Contagion Introduction Geography—allow people to move between contexts: I Locally: standard SIR model with random mixing I discrete time simulation I β = infection probability I γ = recovery probability I P = probability of travel I Movement distance: Pr(d) ∝ exp(−d/ξ) I ξ = typical travel distance Simple disease spreading models Background Prediction References Frame 43/58 Biological Contagion A toy agent-based model Introduction Simple disease spreading models Schematic: Background Prediction References b=2 l=3 n=8 x ij =2 i j Frame 44/58 Model output Biological Contagion Introduction Simple disease spreading models Background Prediction I Define P0 = Expected number of infected individuals leaving initially infected context. I Need P0 > 1 for disease to spread (independent of R0 ). I Limit epidemic size by restricting frequency of travel and/or range References Frame 45/58 Model output Biological Contagion Introduction Simple disease spreading models Background Prediction I Define P0 = Expected number of infected individuals leaving initially infected context. I Need P0 > 1 for disease to spread (independent of R0 ). I Limit epidemic size by restricting frequency of travel and/or range References Frame 45/58 Model output Biological Contagion Introduction Simple disease spreading models Background Prediction I Define P0 = Expected number of infected individuals leaving initially infected context. I Need P0 > 1 for disease to spread (independent of R0 ). I Limit epidemic size by restricting frequency of travel and/or range References Frame 45/58 Model output Biological Contagion Introduction Varying ξ: Simple disease spreading models Background Prediction References I Transition in expected final size based on typical movement distance Frame 46/58 Model output Biological Contagion Introduction Varying ξ: Simple disease spreading models Background Prediction References I Transition in expected final size based on typical movement distance (sensible) Frame 46/58 Model output Varying P0 : Biological Contagion Introduction Simple disease spreading models Background Prediction References I Transition in expected final size based on typical number of infectives leaving first group I Travel advisories: ξ has larger effect than P0 . Frame 47/58 Model output Varying P0 : Biological Contagion Introduction Simple disease spreading models Background Prediction References I Transition in expected final size based on typical number of infectives leaving first group (also sensible) I Travel advisories: ξ has larger effect than P0 . Frame 47/58 Model output Varying P0 : Biological Contagion Introduction Simple disease spreading models Background Prediction References I Transition in expected final size based on typical number of infectives leaving first group (also sensible) I Travel advisories: ξ has larger effect than P0 . Frame 47/58 Biological Contagion Example model output: size distributions Introduction 683 1942 R0=3 R =12 400 N(ψ) N(ψ) 400 300 200 Background 0 Prediction 300 References 200 100 100 0 0 Simple disease spreading models 0.25 0.5 ψ 0.75 1 0 0 0.25 0.5 0.75 1 ψ I Flat distributions are possible for certain ξ and P. I Different R0 ’s may produce similar distributions I Same epidemic sizes may arise from different R0 ’s Frame 48/58 Biological Contagion Example model output: size distributions Introduction 683 1942 R0=3 R =12 400 N(ψ) N(ψ) 400 300 200 Background 0 Prediction 300 References 200 100 100 0 0 Simple disease spreading models 0.25 0.5 ψ 0.75 1 0 0 0.25 0.5 0.75 1 ψ I Flat distributions are possible for certain ξ and P. I Different R0 ’s may produce similar distributions I Same epidemic sizes may arise from different R0 ’s Frame 48/58 Biological Contagion Example model output: size distributions Introduction 683 1942 R0=3 R =12 400 N(ψ) N(ψ) 400 300 200 Background 0 Prediction 300 References 200 100 100 0 0 Simple disease spreading models 0.25 0.5 ψ 0.75 1 0 0 0.25 0.5 0.75 1 ψ I Flat distributions are possible for certain ξ and P. I Different R0 ’s may produce similar distributions I Same epidemic sizes may arise from different R0 ’s Frame 48/58 Biological Contagion Example model output: size distributions Introduction 683 1942 R0=3 R =12 400 N(ψ) N(ψ) 400 300 200 Background 0 Prediction 300 References 200 100 100 0 0 Simple disease spreading models 0.25 0.5 ψ 0.75 1 0 0 0.25 0.5 0.75 1 ψ I Flat distributions are possible for certain ξ and P. I Different R0 ’s may produce similar distributions I Same epidemic sizes may arise from different R0 ’s Frame 48/58 Biological Contagion Example model output: size distributions Introduction 683 1942 R0=3 R =12 400 N(ψ) N(ψ) 400 300 200 Background 0 Prediction 300 References 200 100 100 0 0 Simple disease spreading models 0.25 0.5 ψ 0.75 1 0 0 0.25 0.5 0.75 1 ψ I Flat distributions are possible for certain ξ and P. I Different R0 ’s may produce similar distributions I Same epidemic sizes may arise from different R0 ’s Frame 48/58 Biological Contagion Model output—resurgence Introduction Simple disease spreading models Background Prediction # New cases Standard model: 6000 References D R0=3 4000 2000 0 0 500 1000 1500 t Frame 49/58 Biological Contagion Model output—resurgence Introduction Standard model with transport: Simple disease spreading models # New cases Background 200 Prediction E R0=3 References 100 0 0 500 1000 1500 # New cases t 400 G R0=3 200 0 0 500 1000 1500 t Frame 50/58 The upshot Biological Contagion Introduction Simple disease spreading models Background Multiscale population structure Prediction References Frame 51/58 The upshot Biological Contagion Introduction Simple disease spreading models Background Multiscale population structure + stochasticity Prediction References Frame 51/58 The upshot Biological Contagion Introduction Simple disease spreading models Background Multiscale population structure + stochasticity Prediction References leads to resurgence + broad epidemic size distributions Frame 51/58 Conclusions Biological Contagion Introduction I For this model, epidemic size is highly unpredictable Simple disease spreading models Background Prediction I Model is more complicated than SIR but still simple I We haven’t even included normal social responses such as travel bans and self-quarantine. I The reproduction number R0 is not very useful. R0 , however measured, is not informative about I References 1. how likely the observed epidemic size was, 2. and how likely future epidemics will be. I Problem: R0 summarises one epidemic after the fact and enfolds movement, everything. Frame 52/58 Conclusions Biological Contagion Introduction I For this model, epidemic size is highly unpredictable Simple disease spreading models Background Prediction I Model is more complicated than SIR but still simple I We haven’t even included normal social responses such as travel bans and self-quarantine. I The reproduction number R0 is not very useful. R0 , however measured, is not informative about I References 1. how likely the observed epidemic size was, 2. and how likely future epidemics will be. I Problem: R0 summarises one epidemic after the fact and enfolds movement, everything. Frame 52/58 Conclusions Biological Contagion Introduction I For this model, epidemic size is highly unpredictable Simple disease spreading models Background Prediction I Model is more complicated than SIR but still simple I We haven’t even included normal social responses such as travel bans and self-quarantine. I The reproduction number R0 is not very useful. R0 , however measured, is not informative about I References 1. how likely the observed epidemic size was, 2. and how likely future epidemics will be. I Problem: R0 summarises one epidemic after the fact and enfolds movement, everything. Frame 52/58 Conclusions Biological Contagion Introduction I For this model, epidemic size is highly unpredictable Simple disease spreading models Background Prediction I Model is more complicated than SIR but still simple I We haven’t even included normal social responses such as travel bans and self-quarantine. I The reproduction number R0 is not very useful. R0 , however measured, is not informative about I References 1. how likely the observed epidemic size was, 2. and how likely future epidemics will be. I Problem: R0 summarises one epidemic after the fact and enfolds movement, everything. Frame 52/58 Conclusions Biological Contagion Introduction I For this model, epidemic size is highly unpredictable Simple disease spreading models Background Prediction I Model is more complicated than SIR but still simple I We haven’t even included normal social responses such as travel bans and self-quarantine. I The reproduction number R0 is not very useful. R0 , however measured, is not informative about I References 1. how likely the observed epidemic size was, 2. and how likely future epidemics will be. I Problem: R0 summarises one epidemic after the fact and enfolds movement, everything. Frame 52/58 Conclusions Biological Contagion Introduction I For this model, epidemic size is highly unpredictable Simple disease spreading models Background Prediction I Model is more complicated than SIR but still simple I We haven’t even included normal social responses such as travel bans and self-quarantine. I The reproduction number R0 is not very useful. R0 , however measured, is not informative about I References 1. how likely the observed epidemic size was, 2. and how likely future epidemics will be. I Problem: R0 summarises one epidemic after the fact and enfolds movement, everything. Frame 52/58 Conclusions Biological Contagion Introduction I For this model, epidemic size is highly unpredictable Simple disease spreading models Background Prediction I Model is more complicated than SIR but still simple I We haven’t even included normal social responses such as travel bans and self-quarantine. I The reproduction number R0 is not very useful. R0 , however measured, is not informative about I References 1. how likely the observed epidemic size was, 2. and how likely future epidemics will be. I Problem: R0 summarises one epidemic after the fact and enfolds movement, everything. Frame 52/58 Conclusions Biological Contagion Introduction I For this model, epidemic size is highly unpredictable Simple disease spreading models Background Prediction I Model is more complicated than SIR but still simple I We haven’t even included normal social responses such as travel bans and self-quarantine. I The reproduction number R0 is not very useful. R0 , however measured, is not informative about I References 1. how likely the observed epidemic size was, 2. and how likely future epidemics will be. I Problem: R0 summarises one epidemic after the fact and enfolds movement, everything. Frame 52/58 Conclusions Biological Contagion Introduction Simple disease spreading models Background Prediction I Disease spread highly sensitive to population structure I Rare events may matter enormously I More support for controlling population movement References Frame 53/58 Conclusions Biological Contagion Introduction Simple disease spreading models Background Prediction I Disease spread highly sensitive to population structure I Rare events may matter enormously I More support for controlling population movement References Frame 53/58 Conclusions Biological Contagion Introduction Simple disease spreading models Background Prediction I Disease spread highly sensitive to population structure I Rare events may matter enormously (e.g., an infected individual taking an international flight) I More support for controlling population movement References Frame 53/58 Conclusions Biological Contagion Introduction Simple disease spreading models Background Prediction I Disease spread highly sensitive to population structure I Rare events may matter enormously (e.g., an infected individual taking an international flight) I More support for controlling population movement References Frame 53/58 Conclusions Biological Contagion Introduction Simple disease spreading models Background Prediction I Disease spread highly sensitive to population structure I Rare events may matter enormously (e.g., an infected individual taking an international flight) I More support for controlling population movement (e.g., travel advisories, quarantine) References Frame 53/58 Conclusions Biological Contagion Introduction What to do: Simple disease spreading models Background I Need to separate movement from disease I R0 needs a friend or two. I Need R0 > 1 and P0 > 1 and ξ sufficiently large for disease to have a chance of spreading Prediction References More wondering: I Exactly how important are rare events in disease spreading? I Again, what is N? Frame 54/58 Conclusions Biological Contagion Introduction What to do: Simple disease spreading models Background I Need to separate movement from disease I R0 needs a friend or two. I Need R0 > 1 and P0 > 1 and ξ sufficiently large for disease to have a chance of spreading Prediction References More wondering: I Exactly how important are rare events in disease spreading? I Again, what is N? Frame 54/58 Conclusions Biological Contagion Introduction What to do: Simple disease spreading models Background I Need to separate movement from disease I R0 needs a friend or two. I Need R0 > 1 and P0 > 1 and ξ sufficiently large for disease to have a chance of spreading Prediction References More wondering: I Exactly how important are rare events in disease spreading? I Again, what is N? Frame 54/58 Conclusions Biological Contagion Introduction What to do: Simple disease spreading models Background I Need to separate movement from disease I R0 needs a friend or two. I Need R0 > 1 and P0 > 1 and ξ sufficiently large for disease to have a chance of spreading Prediction References More wondering: I Exactly how important are rare events in disease spreading? I Again, what is N? Frame 54/58 Conclusions Biological Contagion Introduction What to do: Simple disease spreading models Background I Need to separate movement from disease I R0 needs a friend or two. I Need R0 > 1 and P0 > 1 and ξ sufficiently large for disease to have a chance of spreading Prediction References More wondering: I Exactly how important are rare events in disease spreading? I Again, what is N? Frame 54/58 Simple disease spreading models Biological Contagion Introduction Simple disease spreading models Background Prediction Attempts to use beyond disease: I Adoption of ideas/beliefs (Goffman & Newell, 1964) I Spread of rumors (Daley & Kendall, 1965) I Diffusion of innovations (Bass, 1969) I Spread of fanatical behavior (Castillo-Chávez & Song, 2003) References Frame 55/58 Simple disease spreading models Biological Contagion Introduction Simple disease spreading models Background Prediction Attempts to use beyond disease: I Adoption of ideas/beliefs (Goffman & Newell, 1964) I Spread of rumors (Daley & Kendall, 1965) I Diffusion of innovations (Bass, 1969) I Spread of fanatical behavior (Castillo-Chávez & Song, 2003) References Frame 55/58 Simple disease spreading models Biological Contagion Introduction Simple disease spreading models Background Prediction Attempts to use beyond disease: I Adoption of ideas/beliefs (Goffman & Newell, 1964) I Spread of rumors (Daley & Kendall, 1965) I Diffusion of innovations (Bass, 1969) I Spread of fanatical behavior (Castillo-Chávez & Song, 2003) References Frame 55/58 Simple disease spreading models Biological Contagion Introduction Simple disease spreading models Background Prediction Attempts to use beyond disease: I Adoption of ideas/beliefs (Goffman & Newell, 1964) I Spread of rumors (Daley & Kendall, 1965) I Diffusion of innovations (Bass, 1969) I Spread of fanatical behavior (Castillo-Chávez & Song, 2003) References Frame 55/58 Simple disease spreading models Biological Contagion Introduction Simple disease spreading models Background Prediction Attempts to use beyond disease: I Adoption of ideas/beliefs (Goffman & Newell, 1964) I Spread of rumors (Daley & Kendall, 1965) I Diffusion of innovations (Bass, 1969) I Spread of fanatical behavior (Castillo-Chávez & Song, 2003) References Frame 55/58 References I Biological Contagion Introduction P. M. Blau and J. E. Schwartz. Crosscutting Social Circles. Academic Press, Orlando, FL, 1984. Simple disease spreading models Background Prediction References R. L. Breiger. The duality of persons and groups. Social Forces, 53(2):181–190, 1974. E. Hoffer. The True Believer: On The Nature Of Mass Movements. Harper and Row, New York, 1951. E. Hoffer. The Passionate State of Mind: And Other Aphorisms. Buccaneer Books, 1954. Frame 56/58 References II W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A, 115:700–721, 1927. pdf () Biological Contagion Introduction Simple disease spreading models Background Prediction References W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. III. Further studies of the problem of endemicity. Proc. R. Soc. Lond. A, 141(843):94–122, 1927. pdf () W. O. Kermack and A. G. McKendrick. Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc. R. Soc. Lond. A, 138(834):55–83, 1927. pdf () Frame 57/58 References III J. D. Murray. Mathematical Biology. Springer, New York, Third edition, 2002. C. J. Rhodes and R. M. Anderson. Power laws governing epidemics in isolated populations. Nature, 381:600–602, 1996. pdf () Biological Contagion Introduction Simple disease spreading models Background Prediction References G. Simmel. The number of members as determining the sociological form of the group. I. American Journal of Sociology, 8:1–46, 1902. D. J. Watts, P. S. Dodds, and M. E. J. Newman. Identity and search in social networks. Science, 296:1302–1305, 2002. pdf () Frame 58/58
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