Mathematical modelling of infectious disease

From Wikipedia, the free encyclopedia

Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. Models use some basic assumptions and mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programmes. The modelling can help in deciding which intervention/s to avoid and which to trial.

History

The modeling of infectious diseases is a tool which has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic.

The first scientist who systematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt’s analysis of causes of death is considered the beginning of the “theory of competing risks” which according to Daley and Gani  is “a theory that is now well established among modern epidemiologists”. 

The earliest account of mathematical modelling of spread of disease was carried out in 1766 by Daniel Bernoulli. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox. The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy from 26 years 7 months to 29 years 9 months. Daniel Bernoulli's work preceded the modern understanding of germ theory. 

In the early 20th century, William Hamer and Ronald Ross applied the law of mass action to explain epidemic behavior. 

The 1920s saw the emergence of compartmental models. The Kermack–McKendrick epidemic model (1927) and the Reed–Frost epidemic model (1928) both describe the relationship between susceptible, infected and immune individuals in a population. The Kermack–McKendrick epidemic model model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics.

Assumptions

Models are only as good as the assumptions on which they are based. If a model makes predictions which are out of line with observed results and the mathematics is correct, the initial assumptions must change to make the model useful.

• Rectangular and stationary age distribution, i.e., everybody in the population lives to age L and then dies, and for each age (up to L) there is the same number of people in the population. This is often well-justified for developed countries where there is a low infant mortality and much of the population lives to the life expectancy.
• Homogeneous mixing of the population, i.e., individuals of the population under scrutiny assort and make contact at random and do not mix mostly in a smaller subgroup. This assumption is rarely justified because social structure is widespread. For example, most people in London only make contact with other Londoners. Further, within London then there are smaller subgroups, such as the Turkish community or teenagers (just to give two examples), who mix with each other more than people outside their group. However, homogeneous mixing is a standard assumption to make the mathematics tractable.

Types of epidemic models

Stochastic

"Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. Stochastic models depend on the chance variations in risk of exposure, disease and other illness dynamics.

Deterministic

When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. In a deterministic model, individuals in the population are assigned to different subgroups or compartments, each representing a specific stage of the epidemic. Letters such as M, S, E, I, and R are often used to represent different stages. 

The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model.

Reproduction number

The basic reproduction number (denoted by R₀) is a measure of how transferrable a disease is. It is the average number of people that a single infectious person will infect over the course of their infection. This quantity determines whether the infection will spread exponentially, die out, or remain constant: if R₀ > 1, then each person on average infects more than one other person so the disease will spread; if R₀ < 1, then each person infects less than one person on average so the disease will die out; and if R₀ = 1, then each person will infect exactly one other person, so the disease will become endemic: it will move throughout the population but not increase or decrease. 

The basic reproduction number can be computed as a ratio of known rates over time: if an infectious individual contacts β other people per unit time, if all of those people are assumed to contract the disease, and if the disease has a mean infectious period of 1/γ, then the basic reproduction number is just R₀ = β/γ. Some diseases have multiple possible latency periods, in which case the reproduction number for the disease overall is the sum of the reproduction number for each transition time into the disease. For example, Blower et al. model two forms of tuberculosis infection: in the fast case, the symptoms show up immediately after exposure; in the slow case, the symptoms develop years after the initial exposure (endogenous reactivation). The overall reproduction number is the sum of the two forms of contraction: R₀ = R₀^FAST + R₀^SLOW.

Endemic steady state

An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow exponentially and there will be an epidemic, any less and the disease will die out). In mathematical terms, that is:

${\displaystyle \ R_{0}\ =1.}$

The basic reproduction number (R₀) of the disease, assuming everyone is susceptible, multiplied by the proportion of the population that is actually susceptible (S) must be one (since those who are not susceptible do not feature in our calculations as they cannot contract the disease). Notice that this relation means that for a disease to be in the endemic steady state, the higher the basic reproduction number, the lower the proportion of the population susceptible must be, and vice versa.

Assume the rectangular stationary age distribution and let also the ages of infection have the same distribution for each birth year. Let the average age of infection be A, for instance when individuals younger than A are susceptible and those older than A are immune (or infectious). Then it can be shown by an easy argument that the proportion of the population that is susceptible is given by:

${\displaystyle S={\frac {A}{L}}.}$

But the mathematical definition of the endemic steady state can be rearranged to give:

${\displaystyle S={\frac {1}{R_{0}}}.}$

Therefore, due to the transitive property:

${\displaystyle {\frac {1}{R_{0}}}={\frac {A}{L}}\Rightarrow R_{0}={\frac {L}{A}}.}$

This provides a simple way to estimate the parameter R₀ using easily available data. 

For a population with an exponential age distribution,

${\displaystyle R_{0}=1+{\frac {L}{A}}.}$

This allows for the basic reproduction number of a disease given A and L in either type of population distribution.

Modelling epidemics

The SIR model is one of the more basic models used for modelling epidemics. There are a large number of modifications to the model.

The SIR model

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments: susceptible, ${\displaystyle S(t)}$; infected, ${\displaystyle I(t)}$; and removed, ${\displaystyle R(t)}$. The compartments used for this model consist of three classes:

• ${\displaystyle S(t)}$ is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease.
• ${\displaystyle I(t)}$ denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category.
• ${\displaystyle R(t)}$ is the compartment used for those individuals who have been infected and then removed from the disease, either due to immunization or due to death. Those in this category are not able to be infected again or to transmit the infection to others.

The flow of this model may be considered as follows:

${\displaystyle {\color {blue}{{\mathcal {S}}\rightarrow {\mathcal {I}}\rightarrow {\mathcal {R}}}}}$

Using a fixed population, ${\displaystyle N=S(t)+I(t)+R(t)}$, Kermack and McKendrick derived the following equations:

${\displaystyle {\frac {dS}{dt}}=-{\frac {\beta SI}{N}}}$

${\displaystyle {\frac {dI}{dt}}={\frac {\beta SI}{N}}-\gamma I}$

${\displaystyle {\frac {dR}{dt}}=\gamma I}$

Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of ${\displaystyle \beta }$, which is considered the contact or infection rate of the disease. Therefore, an infected individual makes contact and is able to transmit the disease with ${\displaystyle \beta N}$ others per unit time and the fraction of contacts by an infected with a susceptible is ${\displaystyle S/N}$. The number of new infections in unit time per infective then is ${\displaystyle \beta N(S/N)}$, giving the rate of new infections (or those leaving the susceptible category) as ${\displaystyle \beta N(S/N)I=\beta SI}$. For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, a number equal to the fraction (${\displaystyle \gamma }$ which represents the mean recovery/death rate, or ${\displaystyle 1/\gamma }$ the mean infective period) of infectives are leaving this class per unit time to enter the removed class. These processes which occur simultaneously are referred to as the Law of Mass Action, a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned. Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.

Steady state solutions

The expected duration of susceptibility will be${\textstyle E[min(T_{L}|T_{S})]}$where ${\textstyle T_{L}}$reflects the time alive (life expectancy) and ${\textstyle T_{S}}$reflects the time in the susceptible state before becoming infected, which can be simplified to: 

${\displaystyle E[min(T_{L}|T_{S})]=\int _{0}^{\inf[}exp^{-(\mu +\delta )}={\frac {1}{\mu +\delta }}}$,

such that the number of susceptible persons is the number entering the susceptible compartment ${\textstyle \mu N}$times the duration of susceptibility: 

${\displaystyle S={\frac {\mu N}{\mu +\lambda }}}$.

Analogously, the steady-state number of infected persons is the number entering the infected state from the susceptible state (number susceptible, times rate of infection ${\textstyle \lambda ={\frac {\beta I}{N}}}$, times the duration of infectiousness ${\textstyle {\frac {1}{\mu +v}}}$: 

${\displaystyle I={\frac {\mu N}{\mu +\lambda }}\lambda {\frac {1}{\mu +v}}}$.

Other compartmental models

There are a large number of modifications of the SIR model, including those that include births and deaths, where upon recovery there is no immunity (SIS model), where immunity lasts only for a short period of time (SIRS), where there is a latent period of the disease where the person is not infectious (SEIS and SEIR), and where infants can be born with immunity (MSIR).

Infectious disease dynamics

Mathematical models need to integrate the increasing volume of data being generated on host-pathogen interactions. Many theoretical studies of the population dynamics, structure and evolution of infectious diseases of plants and animals, including humans, are concerned with this problem.

Research topics include:

• transmission, spread and control of infection
• epidemiological networks
• spatial epidemiology
• persistence of pathogens within hosts
• intra-host dynamics
• immuno-epidemiology
• virulence
• Strain (biology) structure and interactions
• antigenic shift
• phylodynamics
• pathogen population genetics
• evolution and spread of resistance
• role of host genetic factors
• statistical and mathematical tools and innovations
• role and identification of infection reservoirs

Mathematics of mass vaccination

If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population. Thus, if this level can be exceeded by vaccination, the disease can be eliminated. An example of this being successfully achieved worldwide is the global smallpox eradication, with the last wild case in 1977. The WHO is carrying out a similar vaccination campaign to eradicate polio.

The herd immunity level will be denoted q. Recall that, for a stable state:

${\displaystyle \ R_{0}\cdot S=1.}$

 

In turn,

 

${\displaystyle R_{0}={\frac {N}{S}}={\frac {\mu NE(T_{L})}{\mu NE[min(T_{L},T_{S})}}={\frac {E(T_{L})}{E[min(T_{L},T_{S})]}}}$,

 

which is approximately:

 

${\displaystyle {\frac {E(T_{L})}{E(T_{S})}}=1+{\frac {\lambda }{\mu }}={\frac {\beta N}{v}}}$.

S will be (1 − q), since q is the proportion of the population that is immune and q + S must equal one (since in this simplified model, everyone is either susceptible or immune). Then:

${\displaystyle \ R_{0}\cdot (1-q)=1,}$

${\displaystyle 1-q={\frac {1}{R_{0}}},}$

${\displaystyle q=1-{\frac {1}{R_{0}}}.}$

Remember that this is the threshold level. If the proportion of immune individuals exceeds this level due to a mass vaccination programme, the disease will die out. 

We have just calculated the critical immunisation threshold (denoted q_c). It is the minimum proportion of the population that must be immunised at birth (or close to birth) in order for the infection to die out in the population.

${\displaystyle q_{c}=1-{\frac {1}{R_{0}}}}$.

 

Because the fraction of the final size of the population p that is never infected can be defined as:

 

${\displaystyle \lim _{t->\inf }S(t)=\exp ^{-\int _{0}^{\inf }{\lambda (t)dt}}=1-p}$.

 

Hence,

 

${\displaystyle p=1-\exp ^{-\int _{0}^{\inf }{\beta I(t)dt}}=1-\exp ^{-R_{0}p}}$.

 

Solving for ${\displaystyle \ R_{0}}$, we obtain:

 

${\displaystyle {R_{0}}={\frac {-ln(1-p)}{p}}}$.

When mass vaccination cannot exceed the herd immunity

If the vaccine used is insufficiently effective or the required coverage cannot be reached (for example due to popular resistance), the programme may fail to exceed q_c. Such a programme can, however, disturb the balance of the infection without eliminating it, often causing unforeseen problems. 

Suppose that a proportion of the population q (where q < q_c) is immunised at birth against an infection with R₀>1. The vaccination programme changes R₀ to R_q where

${\displaystyle \ R_{q}=R_{0}(1-q)}$

This change occurs simply because there are now fewer susceptibles in the population who can be infected. R_q is simply R₀ minus those that would normally be infected but that cannot be now since they are immune. 

As a consequence of this lower basic reproduction number, the average age of infection A will also change to some new value A_q in those who have been left unvaccinated. 

Recall the relation that linked R₀, A and L. Assuming that life expectancy has not changed, now:

${\displaystyle \ R_{q}={\frac {L}{A_{q}}},}$

${\displaystyle \ A_{q}={\frac {L}{R_{q}}}={\frac {L}{R_{0}(1-q)}}.}$

But R₀ = L/A so:

${\displaystyle \ {A_{q}}={\frac {L}{(L/A)(1-q)}}={\frac {AL}{L(1-q)}}={\frac {A}{1-q}}.}$

Thus the vaccination programme will raise the average age of infection, another mathematical justification for a result that might have been intuitively obvious. Unvaccinated individuals now experience a reduced force of infection due to the presence of the vaccinated group. 

However, it is important to consider this effect when vaccinating against diseases that are more severe in older people. A vaccination programme against such a disease that does not exceed q_c may cause more deaths and complications than there were before the programme was brought into force as individuals will be catching the disease later in life. These unforeseen outcomes of a vaccination programme are called perverse effects.

When mass vaccination exceeds the herd immunity

If a vaccination programme causes the proportion of immune individuals in a population to exceed the critical threshold for a significant length of time, transmission of the infectious disease in that population will stop. This is known as elimination of the infection and is different from eradication.

Elimination

Interruption of endemic transmission of an infectious disease, which occurs if each infected individual infects less than one other, is achieved by maintaining vaccination coverage to keep the proportion of immune individuals above the critical immunisation threshold.

Eradication

Reduction of infective organisms in the wild worldwide to zero. So far, this has only been achieved for smallpox and rinderpest. To get to eradication, elimination in all world regions must be achieved.

Epidemiology

From Wikipedia, the free encyclopedia

Epidemiology is the study and analysis of the distribution (who, when, and where) and determinants of health and disease conditions in defined populations. 

 

It is the cornerstone of public health, and shapes policy decisions and evidence-based practice by identifying risk factors for disease and targets for preventive healthcare. Epidemiologists help with study design, collection, and statistical analysis of data, amend interpretation and dissemination of results (including peer review and occasional systematic review). Epidemiology has helped develop methodology used in clinical research, public health studies, and, to a lesser extent, basic research in the biological sciences.

Major areas of epidemiological study include disease causation, transmission, outbreak investigation, disease surveillance, environmental epidemiology, forensic epidemiology, occupational epidemiology, screening, biomonitoring, and comparisons of treatment effects such as in clinical trials. Epidemiologists rely on other scientific disciplines like 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, and engineering for exposure assessment.

Etymology

Epidemiology, literally meaning "the study of what is upon the people", is derived from Greek, Modern epi, meaning 'upon, among', demos, meaning 'people, district', and logos, meaning 'study, word, discourse', suggesting that it applies only to human populations. However, the term is widely used in studies of zoological populations (veterinary epidemiology), although the term "epizoology" is available, and it has also been applied to studies of plant populations (botanical or plant disease epidemiology).

The distinction between "epidemic" and "endemic" was first drawn by Hippocrates, to distinguish between diseases that are "visited upon" a population (epidemic) from those that "reside within" a population (endemic). The term "epidemiology" appears to have first been used to describe the study of epidemics in 1802 by the Spanish physician Villalba in Epidemiología Española. Epidemiologists also study the interaction of diseases in a population, a condition known as a syndemic. 

The term epidemiology is now widely applied to cover the description and causation of not only epidemic disease, but of disease in general, and even many non-disease, health-related conditions, such as high blood pressure and obesity. Therefore, this epidemiology is based upon how the pattern of the disease causes change in the function of everyone.

History

The Greek physician Hippocrates, known as the father of medicine, sought a logic to sickness; he is the first person known to have examined the relationships between the occurrence of disease and environmental influences. Hippocrates 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).

Modern era

In the middle of the 16th century, a doctor from Verona named Girolamo Fracastoro was the first to propose a theory that these very small, unseeable, particles that cause disease 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). In 1543 he wrote a book De contagione et contagiosis morbis, in which he was the first to promote personal and environmental hygiene to prevent disease. The development of a sufficiently powerful microscope by Antonie van Leeuwenhoek in 1675 provided visual evidence of living particles consistent with a germ theory of disease. 

Wu Youke (1582–1652) 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, of germs as a cause of epidemic diseases (source: http://baike.baidu.com/view/143117.htm). His concepts are still considered in current scientific research in relation to Traditional Chinese Medicine studies (see: http://apps.who.int/medicinedocs/en/d/Js6170e/4.html). 

Another pioneer, Thomas Sydenham (1624–1689), was the first to distinguish the fevers of Londoners in the later 1600s. His theories on cures of fevers met with 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.

Original map by John Snow showing the clusters of cholera cases in the London epidemic of 1854.

John Graunt, a haberdasher and amateur statistician, published Natural and Political Observations ... upon the Bills of Mortality in 1662. In it, he analysed the mortality rolls in London before the Great Plague, presented one of the first life tables, and 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, and is also known as the father of (modern) epidemiology. He began with noticing the significantly higher death rates in two areas supplied by Southwark Company. His identification of the Broad Street pump as the cause of the Soho epidemic is considered the classic example of epidemiology. Snow used chlorine in an attempt to clean the water and removed the handle; this ended the outbreak. This has been perceived as a major event in the history of public health and regarded as the founding event of the science of epidemiology, having helped shape public health policies around the world. However, Snow's research and preventive measures to avoid further outbreaks were not fully accepted or put into practice until after his death. 

Other pioneers include Danish physician Peter Anton Schleisner, who in 1849 related his work on the prevention of the epidemic of neonatal tetanus on the Vestmanna Islands in Iceland. Another important pioneer was Hungarian physician Ignaz Semmelweis, who in 1847 brought down infant mortality at a Vienna hospital by instituting a disinfection procedure. His findings were published in 1850, but 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. Epidemiology research to examine the relationship between these biomarkers analyzed at the molecular level, and disease was broadly named "molecular epidemiology". Specifically, "genetic epidemiology" has been used for epidemiology of germline genetic variation and disease. Genetic variation is typically determined using DNA from peripheral blood leukocytes. Since the 2000s, genome-wide association studies (GWAS) have been commonly performed to identify genetic risk factors for many diseases and health conditions. 

While most molecular epidemiology studies are still using conventional disease diagnosis and classification systems, it is increasingly recognized that disease progression represents inherently heterogeneous processes differing from person to person. Conceptually, each individual has a unique disease process different from any other individual ("the unique disease principle"), considering uniqueness of the exposome (a totality of endogenous and exogenous / environmental exposures) and its unique influence on molecular pathologic process in each individual. Studies to examine the relationship between an exposure and molecular pathologic signature of disease (particularly cancer) became increasingly common throughout the 2000s. However, the use of molecular pathology in epidemiology posed unique challenges including lack of research guidelines and standardized statistical methodologies, and paucity of interdisciplinary experts and training programs. Furthermore, the concept of disease heterogeneity appears to conflict with the long-standing premise in epidemiology that individuals with the same disease name have similar etiologies and disease processes. To resolve these issues and advance population health science in the era of molecular precision medicine, "molecular pathology" and "epidemiology" was integrated to create a new interdisciplinary field of "molecular pathological epidemiology" (MPE), defined as "epidemiology of molecular pathology and heterogeneity of disease". In MPE, investigators analyze the relationships between; (A) environmental, dietary, lifestyle and genetic factors; (B) alterations in cellular or extracellular molecules; and (C) evolution and progression of disease. A better understanding of heterogeneity of disease pathogenesis will further contribute to elucidate etiologies of disease. The MPE approach can be applied to not only neoplastic diseases but also non-neoplastic diseases. The concept and paradigm of MPE have become widespread in the 2010s.

By 2012 it was recognized that many pathogens' evolution is rapid enough to be highly relevant to epidemiology, and that therefore much could be gained from an interdisciplinary approach to infectious disease integrating epidemiology and molecular evolution to "inform control strategies, or even patient treatment."

Types of studies

Epidemiologists employ a range of study designs from the observational to experimental and generally categorized as descriptive, analytic (aiming to further examine known associations or hypothesized relationships), and experimental (a term often equated with clinical or community trials of treatments and other interventions). In observational studies, nature is allowed to "take its course," as epidemiologists observe from the sidelines. Conversely, in experimental studies, the epidemiologist is the one in control of all of the factors entering a certain case study. Epidemiological studies are aimed, where possible, at revealing unbiased relationships between exposures such as alcohol or smoking, biological agents, stress, or chemicals to mortality or morbidity. The identification of causal relationships between these exposures and outcomes is an important aspect of epidemiology. Modern epidemiologists use informatics as a tool.

Observational studies have two components, descriptive and analytical. Descriptive observations pertain to the "who, what, where and when of health-related state occurrence". However, analytical observations deal more with the ‘how’ of a health-related event. Experimental epidemiology contains three case types: randomized controlled trials (often used for new medicine or drug testing), field trials (conducted on those at a high risk of contracting a disease), and community trials (research on social originating diseases).

The term 'epidemiologic triad' is used to describe the intersection of Host, Agent, and Environment in analyzing an outbreak.

Case series

Case-series may refer to the qualitative study of the experience of a single patient, or small group of patients with a similar diagnosis, or to a statistical factor with the potential to produce illness with periods when they are unexposed. 

The former type of study is purely descriptive and cannot be used to make inferences about the general population of patients with that disease. These types of studies, in which an astute clinician identifies an unusual feature of a disease or a patient's history, may lead to a formulation of a new hypothesis. Using the data from the series, analytic studies could be done to investigate possible causal factors. These can include case-control studies or prospective studies. A case-control study would involve matching comparable controls without the disease to the cases in the series. A prospective study would involve following the case series over time to evaluate the disease's natural history.

The latter type, more formally described as self-controlled case-series studies, divide individual patient follow-up time into exposed and unexposed periods and use fixed-effects Poisson regression processes to compare the incidence rate of a given outcome between exposed and unexposed periods. This technique has been extensively used in the study of adverse reactions to vaccination and has been shown in some circumstances to provide statistical power comparable to that available in cohort studies.

Case-control studies

Case-control studies select subjects based on their disease status. It is 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 case-control 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). 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), i.e. OR = (AD/BC). 

	Cases	Controls
Exposed	A	B
Unexposed	C	D

If the OR is significantly greater than 1, then the conclusion is "those with the disease are more likely to have been exposed," whereas 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 (such as recall bias and selection bias). The main challenge is to identify the appropriate control group; the distribution of exposure among the control group should be representative of the distribution in the population that gave rise to the cases. This can be achieved by drawing a random sample from the original population at risk. This has as a consequence that the control group can contain people with the disease under study when the disease has a high attack rate in a population. 

A major drawback for case control studies is that, in order to be considered to be statistically significant, the minimum number of cases required at the 95% confidence interval is related to the odds ratio by the equation:

${\displaystyle {\text{total cases}}=A+C=1.96^{2}(1+N)\left({\frac {1}{\ln(OR)}}\right)^{2}\left({\frac {OR+2{\sqrt {OR}}+1}{\sqrt {OR}}}\right)\approx 15.5(1+N)\left({\frac {1}{\ln(OR)}}\right)^{2}}$

where N is the ratio of cases to controls. As the odds ratio approached 1, approaches 0; rendering case control studies all but useless for low odds ratios. For instance, for an odds ratio of 1.5 and cases = controls, the table shown above would look like this: 

	Cases	Controls
Exposed	103	84
Unexposed	84	103

For an odds ratio of 1.1: 

	Cases	Controls
Exposed	1732	1652
Unexposed	1652	1732

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. However, the point estimate generated is the relative risk (RR), which is the probability of disease for a person in the exposed group, P_e = A / (A + B) over the probability of disease for a person in the unexposed group, P_u = C / (C + D), i.e. RR = P_e / P_u. 

.....	Case	Non-case	Total
Exposed	A	B	(A + B)
Unexposed	C	D	(C + D)

As with the OR, a RR greater than 1 shows association, where the conclusion can be read "those with the exposure were more likely to develop disease." 

Prospective studies have many benefits over case control studies. The RR is a more powerful effect measure than the OR, as the OR is just an estimation of the RR, since true incidence cannot be calculated in a case control study where subjects are selected based on disease status. Temporality can be established in a prospective study, and confounders are more easily controlled for. However, they are more costly, and there is a greater chance of losing subjects to follow-up based on the long time period over which the cohort is followed. 

Cohort studies also are limited by the same equation for number of cases as for cohort studies, but, if the base incidence rate in the study population is very low, the number of cases required is reduced by ½.

Causal inference

Although epidemiology is sometimes viewed as a collection of statistical tools used to elucidate the associations of exposures to health outcomes, a deeper understanding of this science is that of discovering causal relationships. 

"Correlation does not imply causation" is a common theme for much of the epidemiological literature. For epidemiologists, the key is in the term inference. Correlation, or at least association between two variables, is a necessary but not sufficient criteria for inference that one variable causes the other. Epidemiologists use gathered data and a broad range of biomedical and psychosocial theories in an iterative way to generate or expand theory, to test hypotheses, and to make educated, informed assertions about which relationships are causal, and about exactly how they are causal. 

Epidemiologists emphasize that the "one cause – one effect" understanding is a simplistic mis-belief. Most outcomes, whether disease or death, are caused by a chain or web consisting of many component causes. Causes can be distinguished as necessary, sufficient or probabilistic conditions. If a necessary condition can be identified and controlled (e.g., antibodies to a disease agent, energy in an injury), the harmful outcome can be avoided (Robertson, 2015).

Bradford Hill criteria

In 1965, Austin Bradford Hill proposed a series of considerations to help assess evidence of causation, which have come to be commonly known as the "Bradford Hill criteria". In contrast to the explicit intentions of their author, Hill's considerations are now sometimes taught as a checklist to be implemented for assessing causality. Hill himself said "None of my nine viewpoints can bring indisputable evidence for or against the cause-and-effect hypothesis and none can be required sine qua non."

1. Strength of Association: A small association does not mean that there is not a causal effect, though the larger the association, the more likely that it is causal.
2. Consistency of Data: Consistent findings observed by different persons in different places with different samples strengthens the likelihood of an effect.
3. Specificity: Causation is likely if a very specific population at a specific site and disease with no other likely explanation. The more specific an association between a factor and an effect is, the bigger the probability of a causal relationship.
4. Temporality: The effect has to occur after the cause (and if there is an expected delay between the cause and expected effect, then the effect must occur after that delay).
5. Biological gradient: Greater exposure should generally lead to greater incidence of the effect. However, in some cases, the mere presence of the factor can trigger the effect. In other cases, an inverse proportion is observed: greater exposure leads to lower incidence.
6. Plausibility: A plausible mechanism between cause and effect is helpful (but Hill noted that knowledge of the mechanism is limited by current knowledge).
7. Coherence: Coherence between epidemiological and laboratory findings increases the likelihood of an effect. However, Hill noted that "... lack of such [laboratory] evidence cannot nullify the epidemiological effect on associations".
8. Experiment: "Occasionally it is possible to appeal to experimental evidence".
9. Analogy: The effect of similar factors may be considered.

Legal interpretation

Epidemiological studies can only go to prove that an agent could have caused, but not that it did cause, an effect in any particular case:

Epidemiology is concerned with the incidence of disease in populations and does not address the question of the cause of an individual's disease. This question, sometimes referred to as specific causation, is beyond the domain of the science of epidemiology. Epidemiology has its limits at the point where an inference is made that the relationship between an agent and a disease is causal (general causation) and where the magnitude of excess risk attributed to the agent has been determined; that is, epidemiology addresses whether an agent can cause a disease, not whether an agent did cause a specific plaintiff's disease.

In United States law, epidemiology alone cannot prove that a causal association does not exist in general. Conversely, it can be (and is in some circumstances) taken by US courts, in an individual case, to justify an inference that a causal association does exist, based upon a balance of probability. 

The subdiscipline of forensic epidemiology is directed at the investigation of specific causation of disease or injury in individuals or groups of individuals in instances in which causation is disputed or is unclear, for presentation in legal settings.

Population-based health management

Epidemiological practice and the results of epidemiological analysis make a significant contribution to emerging population-based health management frameworks.

Population-based health management encompasses the ability to:

• Assess the health states and health needs of a target population;
• Implement and evaluate interventions that are designed to improve the health of that population; and
• Efficiently and effectively provide care for members of that population in a way that is consistent with the community's cultural, policy and health resource values.

Modern population-based health management is complex, requiring a multiple set of skills (medical, political, technological, mathematical etc.) of which epidemiological practice and analysis is a core component, that is unified with management science to provide efficient and effective health care and health guidance to a population. This task requires the forward looking ability of modern risk management approaches that transform health risk factors, incidence, prevalence and mortality statistics (derived from epidemiological analysis) into management metrics that not only guide how a health system responds to current population health issues, but also how a health system can be managed to better respond to future potential population health issues.

Examples of organizations that use population-based health management that leverage the work and results of epidemiological practice include Canadian Strategy for Cancer Control, Health Canada Tobacco Control Programs, Rick Hansen Foundation, Canadian Tobacco Control Research Initiative.

Each of these organizations use a population-based health management framework called Life at Risk that combines epidemiological quantitative analysis with demographics, health agency operational research and economics to perform:

• Population Life Impacts Simulations: Measurement of the future potential impact of disease upon the population with respect to new disease cases, prevalence, premature death as well as potential years of life lost from disability and death;
• Labour Force Life Impacts Simulations: Measurement of the future potential impact of disease upon the labour force with respect to new disease cases, prevalence, premature death and potential years of life lost from disability and death;
• Economic Impacts of Disease Simulations: Measurement of the future potential impact of disease upon private sector disposable income impacts (wages, corporate profits, private health care costs) and public sector disposable income impacts (personal income tax, corporate income tax, consumption taxes, publicly funded health care costs).

Applied field epidemiology

Applied epidemiology is the practice of using epidemiological methods to protect or improve the health of a population. Applied field epidemiology can include investigating communicable and non-communicable disease outbreaks, mortality and morbidity rates, and nutritional status, among other indicators of health, with the purpose of communicating the results to those who can implement appropriate policies or disease control measures.

Humanitarian context

As the surveillance and reporting of diseases and other health factors becomes increasingly difficult in humanitarian crisis situations, the methodologies used to report the data are compromised. One study found that less than half (42.4%) of nutrition surveys sampled from humanitarian contexts correctly calculated the prevalence of malnutrition and only one-third (35.3%) of the surveys met the criteria for quality. Among the mortality surveys, only 3.2% met the criteria for quality. As nutritional status and mortality rates help indicate the severity of a crisis, the tracking and reporting of these health factors is crucial. 

Vital registries are usually the most effective ways to collect data, but in humanitarian contexts these registries can be non-existent, unreliable, or inaccessible. As such, mortality is often inaccurately measured using either prospective demographic surveillance or retrospective mortality surveys. Prospective demographic surveillance requires lots of manpower and is difficult to implement in a spread-out population. Retrospective morality surveys are prone to selection and reporting biases. Other methods are being developed, but are not common practice yet.

Validity: precision and bias

Different fields in epidemiology have different levels of validity. One way to assess the validity of findings is the ratio of false-positives (claimed effects that are not correct) to false-negatives (studies which fail to support a true effect). To take the field of genetic epidemiology, candidate-gene studies produced over 100 false-positive findings for each false-negative. By contrast genome-wide association appear close to the reverse, with only one false positive for every 100 or more false-negatives. This ratio has improved over time in genetic epidemiology as the field has adopted stringent criteria. By contrast other epidemiological fields have not required such rigorous reporting and are much less reliable as a result.

Random error

Random error is the result of fluctuations around a true value because of sampling variability. Random error is just that: random. It can occur during data collection, coding, transfer, or analysis. Examples of random error include: poorly worded questions, a misunderstanding in interpreting an individual answer from a particular respondent, or a typographical error during coding. Random error affects measurement in a transient, inconsistent manner and it is impossible to correct for random error. 

There is random error in all sampling procedures. This is called sampling error. 

Precision in epidemiological variables is a measure of random error. Precision is also inversely related to random error, so that to reduce random error is to increase precision. Confidence intervals are computed to demonstrate the precision of relative risk estimates. The narrower the confidence interval, the more precise the relative risk estimate. 

There are two basic ways to reduce random error in an epidemiological study. The first is to increase the sample size of the study. In other words, add more subjects to your study. The second is to reduce the variability in measurement in the study. This might be accomplished by using a more precise measuring device or by increasing the number of measurements. 

Note, that if sample size or number of measurements are increased, or a more precise measuring tool is purchased, the costs of the study are usually increased. There is usually an uneasy balance between the need for adequate precision and the practical issue of study cost.

Systematic error

A systematic error or bias occurs when there is a difference between the true value (in the population) and the observed value (in the study) from any cause other than sampling variability. An example of systematic error is if, unknown to you, the pulse oximeter you are using is set incorrectly and adds two points to the true value each time a measurement is taken. The measuring device could be precise but not accurate. Because the error happens in every instance, it is systematic. Conclusions you draw based on that data will still be incorrect. But the error can be reproduced in the future (e.g., by using the same mis-set instrument). 

A mistake in coding that affects all responses for that particular question is another example of a systematic error. 

The validity of a study is dependent on the degree of systematic error. Validity is usually separated into two components:

• Internal validity is dependent on the amount of error in measurements, including exposure, disease, and the associations between these variables. Good internal validity implies a lack of error in measurement and suggests that inferences may be drawn at least as they pertain to the subjects under study.
• External validity pertains to the process of generalizing the findings of the study to the population from which the sample was drawn (or even beyond that population to a more universal statement). This requires an understanding of which conditions are relevant (or irrelevant) to the generalization. Internal validity is clearly a prerequisite for external validity.

Selection bias

Selection bias occurs when study subjects are selected or become part of the study as a result of a third, unmeasured variable which is associated with both the exposure and outcome of interest. For instance, it has repeatedly been noted that cigarette smokers and non smokers tend to differ in their study participation rates. (Sackett D cites the example of Seltzer et al., in which 85% of non smokers and 67% of smokers returned mailed questionnaires.) It is important to note that such a difference in response will not lead to bias if it is not also associated with a systematic difference in outcome between the two response groups.

Information bias

Information bias is bias arising from systematic error in the assessment of a variable. An example of this is recall bias. A typical example is again provided by Sackett in his discussion of a study examining the effect of specific exposures on fetal health: "in questioning mothers whose recent pregnancies had ended in fetal death or malformation (cases) and a matched group of mothers whose pregnancies ended normally (controls) it was found that 28% of the former, but only 20% of the latter, reported exposure to drugs which could not be substantiated either in earlier prospective interviews or in other health records". In this example, recall bias probably occurred as a result of women who had had miscarriages having an apparent tendency to better recall and therefore report previous exposures.

Confounding

Confounding has traditionally been defined as bias arising from the co-occurrence or mixing of effects of extraneous factors, referred to as confounders, with the main effect(s) of interest. A more recent definition of confounding invokes the notion of counterfactual effects. According to this view, when one observes an outcome of interest, say Y=1 (as opposed to Y=0), in a given population A which is entirely exposed (i.e. exposure X = 1 for every unit of the population) the risk of this event will be R_A1. The counterfactual or unobserved risk R_A0 corresponds to the risk which would have been observed if these same individuals had been unexposed (i.e. X = 0 for every unit of the population). The true effect of exposure therefore is: R_A1 − R_A0 (if one is interested in risk differences) or R_A1/R_A0 (if one is interested in relative risk). Since the counterfactual risk R_A0 is unobservable we approximate it using a second population B and we actually measure the following relations: R_A1 − R_B0 or R_A1/R_B0. In this situation, confounding occurs when R_A0 ≠ R_B0. (NB: Example assumes binary outcome and exposure variables.) 

Some epidemiologists prefer to think of confounding separately from common categorizations of bias since, unlike selection and information bias, confounding stems from real causal effects.

The profession

To date, few universities offer epidemiology as a course of study at the undergraduate level. One notable undergraduate program exists at Johns Hopkins University, where students who major in public health can take graduate level courses, including epidemiology, their senior year at the Bloomberg School of Public Health.

Although epidemiologic research is conducted by individuals from diverse disciplines, including clinically trained professionals such as physicians, formal training is available through Masters or Doctoral programs including Master of Public Health (MPH), Master of Science of Epidemiology (MSc.), Doctor of Public Health (DrPH), Doctor of Pharmacy (PharmD), Doctor of Philosophy (PhD), Doctor of Science (ScD). Many other graduate programs, e.g., Doctor of Social Work (DSW), Doctor of Clinical Practice (DClinP), Doctor of Podiatric Medicine (DPM), Doctor of Veterinary Medicine (DVM), Doctor of Nursing Practice (DNP), Doctor of Physical Therapy (DPT), or for clinically trained physicians, Doctor of Medicine (MD) or Bachelor of Medicine and Surgery (MBBS or MBChB) and Doctor of Osteopathic Medicine (DO), include some training in epidemiologic research or related topics, but this training is generally substantially less than offered in training programs focused on epidemiology or public health. Reflecting the strong historical tie between epidemiology and medicine, formal training programs may be set in either schools of public health and medical schools. 

As public health/health protection practitioners, epidemiologists work in a number of different settings. Some epidemiologists work 'in the field'; i.e., in the community, commonly in a public health/health protection service, and are often at the forefront of investigating and combating disease outbreaks. Others work for non-profit organizations, universities, hospitals and larger government entities such as state and local health departments, various Ministries of Health, Doctors without Borders, the Centers for Disease Control and Prevention (CDC), the Health Protection Agency, the World Health Organization (WHO), or the Public Health Agency of Canada. Epidemiologists can also work in for-profit organizations such as pharmaceutical and medical device companies in groups such as market research or clinical development.