Diffusion processes on complex networks

Transcription

1 Diffusion processes on complex networks Lecture 6 - epidemic processes Janusz Szwabiński Outlook: 1. Introduction 2. Classical models of epidemic spreading 3. Basic results from classical epidemiology Further reading: A.-L. Barabasi, Network Science R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, ( file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 1/18

2 Introduction Epidemiology is the study and analysis of the distribution (who, when, and where) and determinants of health and disease conditions in defined populations. In short, trying to work out why certain people are getting ill. the cornerstone of public health shapes policy decisions and evidence-based practice by identifying risk factors for disease and targets for preventive healthcare major areas: disease causation disease transmission outbreak investigation disease surveillance forensic epidemiology occupational epidemiology screening, biomonitoring comparisons of treatment effects relies on: biology - to better understand disease processes statistics - to make efficient use of the data and draw appropriate conclusions social sciences - to better understand proximate and distal causes engineering - exposure assessment mathematics and physics - modeling of complex systems computer sciences - simulations History the Greek physician Hippocrates (the father of medicine) is the first person known to have examined the relationships between the occurrence of disease and environmental influences he believed sickness of the human body to be caused by an imbalance of the four humors (air, fire, water and earth atoms ) the cure to the sickness was to remove or add the humor in question to balance the body this belief led to the application of bloodletting and dieting in medicine he coined the terms endemic (for diseases usually found in some places but not in others) and epidemic (for diseases that are seen at some times but not others) Girolamo Fracastoro (a doctor from Verona, 16th century) was the first to propose a theory that diseases are caused by very small, unseeable, particles that were alive they were considered to be able to spread by air, multiply by themselves and to be destroyable by fire in this way he refuted Galen's miasma theory (poison gas in sick people) he wrote a book De contagione et contagiosis morbis (1543), in which he was the first to promote personal and environmental hygiene to prevent disease the development of a sufficiently powerful microscope (Antonie van Leeuwenhoek, 1675) provided visual evidence of living particles consistent with a germ theory of disease Wu Youke ( ) developed the concept that some diseases were caused by transmissible agents, which he called liqi (pestilential factors) his book Wenyi Lun (Treatise on Acute Epidemic Febrile Diseases) can be regarded as the main etiological work that brought forward the concept, ultimately attributed to Westerners, file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 2/18

3 of germs as a cause of epidemic diseases his concepts are still considered in current scientific research in relation to Traditional Chinese Medicine studies Thomas Sydenham ( ), was the first to distinguish the fevers of Londoners in the later 1600s much resistance from traditional physicians at the time he was not able to find the initial cause of the smallpox fever he researched and treated John Graunt, a haberdasher (pl. sprzedawca artykułów pasmanteryjnych or właściciel sklepu z odzieżą męską) and amateur statistician published Natural and Political Observations... upon the Bills of Mortality in 1662 he analysed the mortality rolls in London before the Great Plague he presented one of the first life tables he reported time trends for many diseases, new and old he provided statistical evidence for many theories on disease, and also refuted some widespread ideas on them John Snow is famous for his investigations into the causes of the 19th century cholera epidemics the father of (modern) epidemiology he began with noticing the significantly higher death rates in two areas supplied by Southwark Company he identified the Broad Street pump as the cause of the Soho epidemic the classic example of epidemiology he used chlorine in an attempt to clean the water and removed the handle - this ended the outbreak perceived as a major event in the history of public health regarded as the founding event of the science of epidemiology Snow s research and preventive measures to avoid further outbreaks were not fully accepted or put into practice until after his death file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 3/18

4 Peter Anton Schleisner (Danish physician) the prevention of the epidemic of neonatal tetanus on the Vestmanna Islands in Iceland (1849) Ignaz Semmelweis (Hungarian physician) brought down infant mortality at a Vienna hospital by instituting a disinfection procedure (1847) his work was ill-received by his colleagues, who discontinued the procedure disinfection did not become widely practiced until British surgeon Joseph Lister 'discovered' antiseptics in 1865 in light of the work of Louis Pasteur in the early 20th century, mathematical methods were introduced into epidemiology by Ronald Ross, Janet Lane-Claypon, Anderson Gray McKendrick, and others another breakthrough was the 1954 publication of the results of a British Doctors Study, led by Richard Doll and Austin Bradford Hill, which lent very strong statistical support to the link between tobacco smoking and lung cancer in the late 20th century, with advancement of biomedical sciences, a number of molecular markers in blood, other biospecimens and environment were identified as predictors of development or risk of a certain disease molecular epidemiology - research to examine the relationship between the biomarkers file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 4/18

5 Types of studies a range of study designs observational descriptive ( who, what, where and when of health-related state occurrence ) analytic (aiming to further examine known associations or hypothesized relationships) in observational studies, nature is allowed to take its course," as epidemiologists observe from the sidelines experimental the epidemiologist is the one in control of all of the factors entering a certain case study Case series the qualitative study of the experience of a single patient, or small group of patients with a similar diagnosis descriptive cannot be used to make inferences about the general population of patients with that disease these types of studies may lead to formulation of a new hypothesis using the data from the series, analytic studies could be done to investigate possible causal factors Case-control studies case-control studies select subjects based on their disease status a retrospective study a group of individuals that are disease positive (the "case" group) is compared with a group of disease negative individuals (the "control" group) the control group should ideally come from the same population that gave rise to the cases the study looks back through time at potential exposures that both groups (cases and controls) may have encountered a 2 2 table is constructed, displaying exposed cases (A), exposed controls (B), unexposed cases (C) and unexposed controls (D) Cases Controls Exposed A B Unexposed C D the statistic generated to measure association is the odds ratio (OR), which is the ratio of the odds of exposure in the cases (A/C) to the odds of exposure in the controls (B/D), AD OR = BC if the OR is significantly greater than 1, then the conclusion is "those with the disease are more likely to have been exposed," if it is close to 1 then the exposure and disease are not likely associated if the OR is far less than one, then this suggests that the exposure is a protective factor in the causation of the disease case-control studies are usually faster and more cost effective than cohort studies, but are sensitive to bias the main challenge is to identify the appropriate control group file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 5/18

6 Cohort studies cohort studies select subjects based on their exposure status the study subjects should be at risk of the outcome under investigation at the beginning of the cohort study this usually means that they should be disease free when the cohort study starts the cohort is followed through time to assess their later outcome status an example of a cohort study would be the investigation of a cohort of smokers and non-smokers over time to estimate the incidence of lung cancer the same 2 2 table is constructed as with the case control study Case Non-case Total Exposed A B (A + B) Unexposed C D (C + D) the point estimate generated is the relative risk (RR), which is the probability of disease for a person in the exposed group, Pe = A/(A + B) over the probability of disease for a person in the unexposed group, Pu = C/(C + D), i.e. RR = Pe Pu a RR greater than 1 shows association, where the conclusion can be read "those with the exposure were more likely to develop disease." RR is a more powerful effect measure than the OR (the OR is just an estimation of the RR) more costly there is a greater chance of losing subjects to follow-up based on the long time period over which the cohort is followed Outbreak investigation Identify the existence of the outbreak (Is the group of ill persons normal for the time of year, geographic area, etc.?) Verify the diagnosis related to the outbreak Create a case definition to define who/what is included as a case Map the spread of the outbreak using Information technology as diagnosis is reported to insurance Develop a hypothesis (What appears to be causing the outbreak?) Study hypotheses (collect data and perform analysis) Refine hypothesis and carry out further study Develop and implement control and prevention systems Release findings to greater communities file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 6/18

7 Classical models of epidemic spreading mathematical models lie at the core of our understanding about infectious diseases Bernoulli's contribution Daniel Bernoulli, a Swiss mathematician and physicist, 1766 first mathematical approach to the spread of a disease used smallpox data from Wrocław (provided by Halley) he wanted to relate the size w(t) of a cohort of individuals at time t after birth with the number of susceptibles x(t) among them who had not been infected with smallpox he assumed that individuals infected by smallpox died immediately or recovered immediately and became immunes z(t), w(t) = x(t) + z(t) he obtained the formula: w(t) x(t) = (1 a) e bt + a where: b - rate of catching smallpox a proportion of those infected who died instantaneously he found that if smallpox could be eliminated, then by age 26 the population would be some 14% larger he used this result to argue the advantages of variolation Variolation (or inoculation) the method first used to immunize an individual against smallpox with material taken from a patient or a recently variolated individual in the hope that a mild but protective infection would result usually carried out by inserting/rubbing powdered smallpox scabs or fluid from pustules into superficial scratches made in the skin first used in China around 100 B.C.!!! Modern modeling Kermack and McKendrick (1927) two important assumptions: compartmentalization an individual is classified based on the stage of the disease affecting him the simplest classification assumess that an individual can be in one of three states or compartments: Susceptible ( S) healthy individuals who have not yet contacted the pathogen Infectious ( I) contagious individuals who have contacted the pathogen and hence can infect others Recovered or Removed ( R) individuals who have been infected before, but have recovered from the disease (or died), hence are not infectious some diseases require additional states like: file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 7/18

8 Immune - individuals who cannot be infected Latent - individuals who have been exposed to the disease, but are not yet contagious individuals can move between compartments homogenous mixing also called fully mixed or mass-action approximation each individual has the same chance of coming into contact with an infected individual it eliminates the need to know the precise contact network on which the disease spreads, replacing it with the assumption that anyone can infect anyone else file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 8/18

9 Susceptible-Infected (SI) model Consider a disease that spreads in a population of N individuals. Denote with S(t) the number of individuals who are susceptible (healthy) at time t and with I(t) the number of individuals that have been already infected. Initially everyone is susceptible and no one is infected, S(0) = N, I(0) = 0 Let β be the likelihood that the disease is transferred from an infected individuals to a healthy one in a unit time. We ask the following question: How many individuals will be infected at some later time t, if a single individual becomes infected at t = 0, I(0) = 1? Within the homogenous mixing hypothesis the probability that an infected person encounters a susceptible individual is Therefore, the average number of infectious individuals in the unit time is Hence, I(t) changes at a rate β S(t) N S(t) I(t)dt N di(t) dt S(t) = β I(t) N Introducing the variables I(t) i(t) =, s(t) = N S(t) N we get file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 9/18

10 di(t) = βs(t)i(t) = βi(t)[1 i(t)] dt s(t) = 1 i(t) The parameter β is called the transmission rate or transmissibility. We solve the last equation for i(t) by writing di i di + = βdt 1 i Integrating both sides we get ln i ln(1 i) + C = βt The initial condition i(t = 0) = i 0 gives C = i 0 1 i 0 Hence i(t) = i 0 e βt 1 i 0 i 0 e βt at the beginning the fraction of infected individuals increases exponentially the characteristic time required to reach a 1/e fraction (i.e. about 36%) of all susceptible individuals is 1 τ = β with time an infected individual encounters fewer and fewer susceptible individuals, hence the growth of i(t) slows down for large t the epidemic ends when everyone has been infected, i.e. i(t) t 1 s(t) t 0 file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 10/18

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12 Susceptible-Infected-Susceptible (SIS) model Most pathogens are eventually defeated by the immune system or by treatment. To capture this fact we need to allow the infected individuals to recover. With that we arrive at the SIS model. It has the same states as the model before, but now infected individuals recover at a fixed rate μ, becoming susceptible again. The equation describing the dynamics of the model is an extension of the one for the SI model, i.e. di(t) dt s(t) = 1 i(t) = βi(t)[1 i(t)] μi(t) with μ being the recovery rate. The μi(t) term captures the rate at which the population recovers from the disease. The solution of the above equation provides the fraction of infected individuals in function of time: i(t) = ( μ 1 β ) Ce (β μ)t 1 + Ce (β μ)t C = i 0 1 i 0 μ β As show before, in the SI model everyone becomes eventually infected. In the SIS model we have 2 possible outcomes: Endemic state for low recovery rate the fraction of infected individuals i(t) follows a logistic curve similar to the one observed for the SI model yet, not everyone is infected, i.e. i(t) reaches a constant value i( ) < 1 We can calculate i( ) by setting di dt = 0 We obtain file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 12/18

13 0 0 0 = βi(1 i) μi = β(1 i) μ = 1 i μ β Hence i( ) = 1 μ β Disease-free state for sufficiently high recovery rate the exponent in the equation for i(t) is negative i dicreases exponentially with time, i.e. initial infection will die out exponentially Basic reproductive number In other words, the SIS model predicts that some pathogens will persist in the population while others die out quickly. To understand what govern the difference between these two outcomes, we write the characteristic time of a pathogen as τ = 1 μ( R 0 1) with R 0 = β μ R 0 is the basic reproductive number file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 13/18

14 it represents the average number of susceptible individuals infected by one during his infectious period in a fully susceptible population i.e. R 0 is the number of new infections each infected individual causes under ideal circumstances valuable for its predictive power if R 0 exceeds unity, τ is positive and the epidemic is in the endemic state each infected individual infects more than one healthy person the pathogen spreads and persists in the population if R 0 < 1 then τ is negative and the epidemic dies out one of the first parameters the epidemiologists estimate for a new pathogen, gauging the severity of the problem they face Disease Transmission R 0 Measles (pl. odra) Airborne Pertusis (pl. krztusiec) Airborne droplet Diphteria (pl. błonica) Saliva 6-7 Smallpox (pl. ospa) Social contact 5-7 Polio Fecal-oral route 5-7 Rubella (pl. różyczka) Airborne droplet 5-7 Mumps (pl. świnka) Airborne droplet 4-7 HIV/AIDS Sexual contact 2-5 SARS Airborne droplet 2-5 Influenza (1918 strain) Airborne droplet 2-3 hi h R f f h h d li h d h file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 14/18

15 Susceptible-Infected-Recovered (SIR) model For many pathogens, like most strains of influenza, individuals develop immunity after they recover from the infection. Hence, instead of returning to the susceptible state, they are "removed" from the population. They do not count any longer from the perspective of the pathogen as they cannot be infected, nor can they infect others. The dynamics of such a process is captured by the SIR model. ds(t) dt di(t) dt dr(t) dt s(t) + i(t) + r(t) = 1 = βi(t)s(t) = μi(t) + βi(t)s(t) = μi(t) file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 15/18

16 Basic results from classical epidemiology the models have different outcomes in general in the early stages on an epidemic their predictions agree with each other depending on the characteristics of the pathogen we need different models to capture the dynamics of an epidemic outbreak file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 16/18

17 Other well-known models Susceptible-Infected-Recovered-Susceptible (SIRS) model temporal immunity (chickenpox) Susceptible-Exposed-Infected-Recovered (SEIR) model influenza-like illnesses exposed means infected but cannot transmit yet Susceptible-Exposed-Infected-Susceptible (SEIS) model file:///home/szwabin/dropbox/zajecia/diffusion/lectures/6_epidemics/6_epidemics.html 17/18

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