Mathematical modelling of infectious diseasesMathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programs. The modelling can help decide which intervention(s) to avoid and which to trial, or can predict future growth patterns, etc.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Basic reproduction numberIn epidemiology, the basic reproduction number, or basic reproductive number (sometimes called basic reproduction ratio or basic reproductive rate), denoted (pronounced R nought or R zero), of an infection is the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection. The definition assumes that no other individuals are infected or immunized (naturally or through vaccination).
Compartmental models in epidemiologyCompartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, or Recovered). People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.
Initial conditionIn mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time.
Mathematical inductionMathematical induction is a method for proving that a statement is true for every natural number , that is, that the infinitely many cases all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). A proof by induction consists of two cases.
Computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
MathematicsMathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them.
Endemic (epidemiology)In epidemiology, an infection is said to be endemic in a specific population or populated place when that infection is constantly present, or maintained at a baseline level, without extra infections being brought into the group as a result of travel or similar means. The term describes the distribution (spread) of an infectious disease among a group of people or within a populated area. An endemic disease always has a steady, predictable number of people getting sick, but that number can be high (hyperendemic) or low (hypoendemic), and the disease can be severe or mild.
List of mathematical jargonThe language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense. Some phrases, like "in general", appear below in more than one section.
Formal proofIn logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system.
Proof by exhaustionProof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: A proof that the set of cases is exhaustive; i.e.
RigourRigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law. "Rigour" comes to English through old French (13th c.
Overlay networkAn overlay network is a computer network that is layered on top of another network. Nodes in the overlay network can be thought of as being connected by virtual or logical links, each of which corresponds to a path, perhaps through many physical links, in the underlying network. For example, distributed systems such as peer-to-peer networks and client–server applications are overlay networks because their nodes run on top of the Internet.
Truth valueIn logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (true or false). In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true.
Animal disease modelAn animal model (short for animal disease model) is a living, non-human, often genetic-engineered animal used during the research and investigation of human disease, for the purpose of better understanding the disease process without the risk of harming a human. Although biological activity in an animal model does not ensure an effect in humans, many drugs, treatments and cures for human diseases are developed in part with the guidance of animal models.
Percolation theoryIn statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles Network theory and Percolation (cognitive psychology).
Computer networkA computer network is a set of computers sharing resources located on or provided by network nodes. Computers use common communication protocols over digital interconnections to communicate with each other. These interconnections are made up of telecommunication network technologies based on physically wired, optical, and wireless radio-frequency methods that may be arranged in a variety of network topologies. The nodes of a computer network can include personal computers, servers, networking hardware, or other specialized or general-purpose hosts.
Percolation thresholdThe percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them.
GoalA goal or objective is an idea of the future or desired result that a person or a group of people envision, plan and commit to achieve. People endeavour to reach goals within a finite time by setting deadlines. A goal is roughly similar to a purpose or aim, the anticipated result which guides reaction, or an end, which is an object, either a physical object or an abstract object, that has intrinsic value. Goal setting Goal-setting theory was formulated based on empirical research and has been called one of the most important theories in organizational psychology.
