Meta tags:

Headings (most frequently used words):

models, of, mass, vaccination, the, infectious, epidemic, compartmental, when, herd, immunity, mathematical, modelling, diseases, contents, history, assumptions, types, sub, exponential, growth, on, networks, reproduction, number, endemic, steady, state, in, epidemiology, continuous, from, seirs, to, density, equation, disease, dynamics, mathematics, reliability, see, also, references, sources, further, reading, external, links, stochastic, deterministic, kinetic, and, mean, field, sir, model, other, cannot, exceed, exceeds,

Text of the page (most frequently used words):
the (233), and (108), #disease (65), model (46), infection (44), models (43), for (41), #infectious (40), population (40), that (38), doi (32), displaystyle (31), mathematical (29), with (28), epidemic (28), edit (28), modelling (25), this (25), from (23), frac (23), are (23), vaccination (21), network (20), pmid (20), diseases (19), epidemiology (19), transmission (19), can (19), will (18), number (17), contact (16), other (16), compartmental (15), individuals (15), immunity (14), basic (14), susceptible (14), needed (14), reproduction (13), pmc (13), then (13), dynamics (13), such (13), infected (13), proportion (13), person (13), modeling (12), health (12), immune (12), have (12), those (12), page (11), vaccine (11), age (11), 2020 (11), bibcode (11), which (11), each (11), used (11), who (10), one (10), herd (10), 978 (10), out (10), mass (10), average (10), been (10), where (10), distribution (9), rate (9), time (9), citation (9), there (9), wikipedia (8), may (8), social (8), based (8), endemic (8), agent (8), mathematics (8), they (8), networks (8), than (8), when (8), not (8), operatorname (8), langle (8), rangle (8), theory (7), eradication (7), respiratory (7), sars (7), control (7), state (7), period (7), isbn (7), science (7), epidemics (7), spread (7), beta (7), only (7), textstyle (7), using (6), was (6), 2025 (6), articles (6), data (6), hepatitis (6), smallpox (6), cov (6), viral (6), pandemic (6), case (6), host (6), links (6), journal (6), level (6), example (6), due (6), has (6), since (6), must (6), die (6), sir (6), people (6), deterministic (6), stochastic (6), assumptions (6), toggle (5), last (5), biology (5), systems (5), simulation (5), clinical (5), vaccines (5), combination (5), research (5), covid (5), program (5), inoculation (5), immunization (5), antimicrobial (5), resistance (5), surveillance (5), community (5), life (5), recovery (5), differential (5), critical (5), size (5), change (5), spreading (5), 1007 (5), now (5), against (5), density (5), growth (5), mean (5), bernoulli (5), also (5), threshold (5), exceeds (5), but (5), cannot (5), types (5), main (5), more (5), move (5), degree (5), between (5), history (5), contents (4), search (4), available (4), terms (4), use (4), july (4), unsourced (4), statements (4), medical (4), retrieved (4), list (4), structure (4), never (4), human (4), virus (4), global (4), like (4), vector (4), outbreak (4), response (4), public (4), curve (4), exposure (4), antibody (4), death (4), course (4), testing (4), factors (4), endogenous (4), individual (4), epidemiological (4), 1038 (4), reviews (4), its (4), 2016 (4), press (4), university (4), hdl (4), new (4), past (4), how (4), 2017 (4), 1016 (4), seirs (4), kermack (4), mckendrick (4), over (4), compartment (4), april (4), arxiv (4), kinetic (4), equations (4), type (4), same (4), infects (4), causes (4), exceed (4), birth (4), changes (4), either (4), min (4), tools (4), equation (4), gamma (4), 997 (4), article (4), steady (4), however (4), any (4), sub (4), interventions (4), tagged (4), random (4), make (4), abms (4), hide (4), sidebar (4), view (3), statistics (3), policy (3), under (3), site (3), inactive (3), january (3), 2022 (3), all (3), short (3), description (3), different (3), software (3), related (3), topics (3), process (3), climate (3), prediction (3), mmr (3), general (3), hib (3), dtap (3), borne (3), group (3), tropical (3), prevention (3), microbial (3), ecology (3), zoonosis (3), emerging (3), genetic (3), antigenic (3), surgical (3), source (3), hygiene (3), tracing (3), post (3), drug (3), measures (3), years (3), latent (3), value (3), methods (3), lower (3), matrix (3), attack (3), generation (3), force (3), open (3), intervention (3), temporal (3), concepts (3), natural (3), risk (3), pathogen (3), world (3), external (3), tool (3), nature (3), book (3), introduction (3), cambridge (3), computational (3), further (3), ross (3), hamer (3), daley (3), gani (3), 1999 (3), brauer (3), 2012 (3), applied (3), link (3), november (3), predicting (3), pandemics (3), 369 (3), influence (3), future (3), 1927 (3), 1103 (3), review (3), mixing (3), modern (3), spatial (3), statistical (3), field (3), game (3), 1111 (3), mortality (3), reliability (3), elimination (3), dies (3), occurs (3), thus (3), unvaccinated (3), populations (3), would (3), expectancy (3), because (3), infty (3), denoted (3), graph (3), interactions (3), evolution (3), many (3), including (3), continuous (3), compartments (3), initial (3), often (3), exponential (3), every (3), means (3), exponentially (3), infect (3), their (3), increase (3), parameters (3), account (3), help (3), subsection (3), languages (2), table (2), safety (2), contacts (2), about (2), privacy (2), non (2), wikimedia (2), foundation (2), commons (2), categories (2), cs1 (2), maint (2), 2023 (2), wikidata (2), theoretical (2), index (2), assessment (2), hydrological (2), transport (2), chemical (2), biological (2), ecosystem (2), scientific (2), phase (2), iii (2), trials (2), 2000 (2), schedule (2), services (2), pox (2), polio (2), salk (2), sabin (2), john (2), dtwp (2), hepb (2), ipv (2), albumin (2), gardasil (2), cervarix (2), hpv (2), bcg (2), malaria (2), fever (2), tick (2), encephalitis (2), flu (2), ebola (2), tuberculosis (2), synthetic (2), classes (2), artificial (2), medicine (2), phylodynamics (2), reverse (2), genomic (2), shift (2), universal (2), precautions (2), travel (2), distancing (2), sanitation (2), sex (2), quarantine (2), food (2), flattening (2), pharmaceutical (2), prophylaxis (2), therapy (2), antiviral (2), stewardship (2), syndrome (2), effect (2), disability (2), outcome (2), carrier (2), asymptomatic (2), secondary (2), pattern (2), states (2), care (2), predictive (2), detection (2), antigen (2), test (2), acid (2), diagnostics (2), susceptibility (2), stain (2), eye (2), blood (2), system (2), genitourinary (2), gastrointestinal (2), tract (2), location (2), geographic (2), soil (2), waifw (2), animal (2), perinatal (2), prenatal (2), wound (2), burn (2), water (2), bite (2), line (2), injection (2), linked (2), aerosol (2), air (2), environment (2), horizontal (2), cross (2), formation (2), biofilm (2), overgrowth (2), heterogeneity (2), load (2), determinants (2), levels (2), status (2), virulence (2), germ (2), lecture (2), vol (2), 540 (2), study (2), 2008 (2), microbiology (2), princeton (2), humans (2), animals (2), reading (2), 1910 (2), oclc (2), 2027 (2), 1929 (2), castillo (2), chavez (2), 4614 (2), barabási (2), sources (2), cite (2), engineering (2), guide (2), reliable (2), were (2), bioinformatics (2), reveals (2), potential (2), 020 (2), principles (2), 1126 (2), 2020sci (2), idm (2), refractory (2), approach (2), jstor (2), september (2), pdf (2), space (2), b978 (2), 044313 (2), physreve (2), 105 (2), physical (2), effects (2), heterogeneous (2), 2015 (2), 925 (2), 2007 (2), 036113 (2), second (2), 368 (2), effective (2), recent (2), cases (2), 030 (2), volume (2), 2982521 (2), royal (2), society (2), nakamura (2), cardoso (2), martinez (2), 191504 (2), dissemination (2), practice (2), m1567 (2), bmj (2), action (2), 580 (2), special (2), 1760 (2), 802260 (2), analysis (2), references (2), see (2), multiple (2), outcomes (2), single (2), shown (2), everywhere (2), less (2), coverage (2), above (2), experience (2), vaccinated (2), leads (2), older (2), occur (2), recall (2), relation (2), assuming (2), simply (2), fewer (2), programme (2), int (2), hence (2), lim (2), lambda (2), defined (2), just (2), calculated (2), aligned (2), cdot (2), 6pt (2), everyone (2), wild (2), after (2), reported (2), role (2), within (2), include (2), need (2), increasing (2), being (2), plants (2), describe (2), ordinary (2), deaths (2), upon (2), category (2), transmit (2), others (2), created (2), considered (2), three (2), animation (2), values (2), rates (2), formulated (2), simple (2), given (2), assumed (2), give (2), rectangular (2), stationary (2), let (2), calculations (2), inputs (2), exactly (2), find (2), dfrac (2), modeled (2), represented (2), mathematically (2), top (2), another (2), tree (2), tag (2), these (2), class (2), large (2), subgroups (2), homogeneous (2), mix (2), smaller (2), assumption (2), justified (2), london (2), lives (2), well (2), observed (2), useful (2), inform (2), months (2), daniel (2), bills (2), graunt (2), predict (2), appearance (2), upload (2), file (2), read (2), log (2), create (2), donate (2), menu (2), add, topic, mobile, cookie, statement, developers, code, conduct, legal, disclaimers, text, additional, apply, you, agree, registered, trademark, profit, organization, inc, creative, attribution, sharealike, license, rendered, parsoid, edited, august, 2026, utc, hidden, 2024, march, needing, citations, https, org, php, title, mathematical_modelling_of_infectious_diseases, oldid, 1372391575, visual, analytics, thinking, computer, visualization, input, output, economic, crime, mapping, construction, management, catastrophe, business, biopsychosocial, integrated, energy, sustainability, wildfire, modular, ocean, groundwater, geologic, atmospheric, environmental, protein, metabolic, cellular, market, withdrawn, vaccinate, your, family, simpsonwood, cdc, conference, alternative, cedillo, secretary, sids, thiomersal, party, oral, aids, hypothesis, ncvia, autism, fraud, lancet, controversy, drew, weissman, max, theiler, jonas, albert, stanley, plotkin, louis, pasteur, paul, offit, edwin, herman, lennette, hilary, koprowski, katalin, karikó, edward, jenner, peter, hotez, maurice, hilleman, franklin, enders, fred, clark, inventors, researchers, hexavalent, nic, ovandrotone, fecundin, nicvax, prostvac, alvac, cea, cancer, androvax, androstenedione, schistosomiasis, hookworm, helminthiasis, trypanosomiasis, rts, protozoan, zika, hiv, herpes, simplex, epstein, barr, dengue, cytomegalovirus, mmrv, influenza, yellow, varicella, zoster, shingles, chicken, syncytial, rsv, valneva, sputnik, skycovione, sinopharm, bibp, sanofi, gsk, pfizer, biontech, oxford, astrazeneca, novavax, moderna, kostaive, janssen, epivaccorona, coronavac, cansino, bharat, biotech, corbevax, rubella, rotavirus, rabies, mumps, mpox, measles, japanese, hantavirus, laiv, h5n8, h5n1, pandemrix, h1n1, rvsv, zebov, chikungunya, adenovirus, shigellosis, streptococcal, clostridioides, difficile, tdap, dpt, typhus, vicps, ty21a, typhoid, tetanus, ppsv, pcv, pneumococcal, plague, pertussis, nmvac4, 135, menzb, meningococcus, lyme, leptospirosis, diphtheria, cholera, brucellosis, anthrax, bacterial, vsd, vaers, children, court, acip, injury, wastage, sage, nitag, gavi, alliance, administration, toxoid, therapeutic, mrna, dna, particle, peptide, subunit, component, live, heterologous, attenuated, inactivated, conjugate, timeline, storage, ingredients, adjuvants, development, j07, induction, bioterrorism, informatics, specialty, economics, discovery, selection, pressure, reassortment, phylogenetics, initiatives, evolutionary, emergent, crispr, biosecurity, ears, net, drift, infections, zoning, wastewater, restrictions, sterilization, screening, safe, ppe, mask, n95, respirator, education, communication, protective, sequestration, notification, lockdown, barrier, nursing, isolation, gloves, hand, washing, disinfection, cordon, sanitaire, ring, preventable, hesitancy, booster, efficacy, effectiveness, repurposed, drugs, pre, phage, monoclonal, immunotherapy, asepsis, antiseptic, ascaricide, anthelmintic, antifungal, prophylactic, antibiotic, sequela, fatigue, late, healing, adjusted, cure, convalescence, silent, subclinical, resolution, relapse, remission, persistent, fulminant, chronic, subacute, acute, reinfection, superinfection, coinfection, primary, biomarker, diagnosis, point, negative, positive, sensitivity, specificity, diagnostic, seroconversion, titer, immunoassay, serology, indirect, rapid, sequencing, pcr, nucleic, amplification, tests, molecular, microbiological, culture, fast, gram, staining, smear, histopathology, microscopy, direct, fetus, mouth, tooth, systemic, generalized, cardiovascular, joint, bone, soft, tissue, skin, nervous, intestinal, chest, ear, nose, throat, upper, anatomical, twindemic, syndemic, sporadic, seasonality, prevalence, mesoendemic, inequality, incidence, hyperendemic, holoendemic, virgin, farr, laws, cluster, occurrence, serial, interval, multiplicity, machine, learning, ocular, mucosal, membrane, cervico, vaginal, trans, placental, fomite, cutaneous, feces, breastmilk, contamination, mosquito, postoperative, percutaneous, insect, intravenous, vascular, droplet, dental, generating, procedure, bioaerosol, routes, sapronosis, breakthrough, generational, difference, neonatal, vertical, congenital, iatrogenic, nosocomial, hospital, contagious, spillover, species, exogenous, seeding, translocation, endogenization, reactivation, normal, flora, modes, super, spreader, window, incubation, shedding, reservoir, focus, chain, war, conflict, urbanization, poverty, poultry, livestock, disaster, flood, humidity, deforestation, commerce, zones, niño, biodiversity, loss, quality, access, stress, pregnancy, behavioral, lifestyle, smoking, predisposition, nutrition, gender, opportunistic, microbiome, cytokine, storm, immunopathology, immunosuppression, immunodeficiency, interaction, diabetes, comorbidity, tropism, multidrug, resistant, bacteria, gene, transfer, fatality, exotoxin, endotoxin, quorum, sensing, pathogenicity, pathogenesis, dose, infectivity, outline, package, spatio, monitoring, phenomena, framework, through, eclipse, stem, enables, across, gleamviz, simulator, interactive, gui, build, simulate, analyze, ode, builder, notes, biomathematics, 1993, 56526, 70514, structures, boily, mâsse, jul, aug, 1997, 9336095, 6990198, bf03404793, 255, canadian, precious, sexually, transmitted, grassly, fraser, june, 18533288, 7097581, nrmicro1845, 477, introductory, applications, vynnycky, white, archived, original, von, csefalvay, elsevier, academic, keeling, rohani, dutton, 610268760, uc2, ark, 13960, t02z1ds0q, macmillan, 609575950, mdp, 39015006657475, old, 521, 01467, texts, 1685, 1686, 107, 07626, costris, vas, schwartz, smith, 7518, 33378907, 3934, mbe, 2020383, 7502, biosciences, renz, widerspick, dräger, december, supplement_2, i821, 33381848, 7773487, 1093, btaa813, i813, fba, guanylate, kinase, target, therapies, pollard, bijker, 2021, 100, 33353987, 7754704, s41577, 00479, immunology, vaccinology, developments, britton, ball, trapman, 6505, 849, 32576668, 7331793, abc6810, 846b, 846, 127, 29928732, 6001967, 001, 113, present, chizhov, pujo, menjouet, schwalger, sensi, 801, 40213022, 11982974, 2468, 0427, issn, 004, 787, multi, scale, 1991, 2059741, 94815, bf02464423, 1927rspsa, 115, 700k, bulletin, contributions, cosma, shalizi, 2018, carnegie, mellon, miller, 2003, burden, 107258, 012107258, 50005, rizi, faqeeh, badie, modiri, kivelä, 35590624, 2022phrve, 105d4313r, 2103, 12634, digital, failures, pastor, satorras, castellano, van, mieghem, vespignani, 979, revmodphys, 2015rvmp, 925p, 1408, 2701, physics, processes, complex, kenah, robins, 17930312, 2215389, 2007phrve, 76c6113k, bio, 0610057, look, maier, brockmann, 6492, 746, 32269067, 7164388, abb4557, 742m, 742, containment, explains, subexponential, confirmed, china, albi, bertaglia, boscheri, dimarco, toscani, uncertain, multiscale, technology, 108, 96561, 96562, 4_3, globally, connected, pareschi, dietz, 1967, rumours, survey, 528, 2307, 505, 130, series, february, 32257317, 7062106, 1098, rsos, 2020rsos, 791504n, improved, uncertainties, autocorrelation, functions, monteiro, 40885, 28106086, 5247741, srep40885, 2017natsr, 740885n, 1606, 07825, reports, efficient, method, comprehensive, computation, tembine, 10419, 257469, 3390, g11040051, games, driven, perspective, huppert, katriel, 2013, 1005, 24266045, 1469, 0691, 12308, 999, maziarz, zach, october, 1360, 32820573, 7461315, jep, 13459, 1352, evaluation, methodological, appraisal, sridhar, majumder, 32317328, 1136, squazzoni, polhill, edmonds, ahrweiler, antosz, scholz, 10037, 19057, 18564, jasss, 4298, societies, matter, during, call, adam, 7803, 318, 32242115, d41586, 01003, 2020natur, 316a, 316, report, simulations, driving, eisinger, thulke, 423, 18784795, 2326892, 1365, 2664, 01439, 2008japec, 415e, 415, facilitates, foppa, pioneer, epidemiologic, 00001, historical, blower, 2004, attempt, caused, advantages, prevent, 15334536, 1002, rmv, 443, 275, virology, hethcote, 653, 1137, s0036144500371907, 2000siamr, 599h, 599, siam, sexual, factor, next, landscape, advantage, examining, simultaneously, rather, making, forecast, broad, degrees, mers, swine, rinderpest, interruption, achieved, maintaining, keep, significant, length, stop, lead, raise, reduced, presence, greater, severity, relatively, later, absence, changed, consequence, some, left, susceptibles, minus, normally, suppose, immunised, insufficiently, required, reached, fail, protect, solving, obtain, fraction, final, minimum, immunized, close, order, remember, begin, end, equal, simplified, selected, approximately, turn, stable, longer, persist, eliminated, enough, prior, 1977, certification, indigenous, 2019, poliovirus, strain, innovations, identification, reservoirs, persistence, pathogens, hosts, genetics, intra, immuno, integrate, generated, studies, concerned, problem, whereas, consider, discrete, semi, infinite, passed, interestingly, derived, interplay, neurons, neuroscience, modifications, births, sis, lasts, sirs, infants, born, msir, seis, seir, removed, able, again, denotes, capable, fixed, recovered, consist, shows, reducing, possible, reduce, referred, appropriate, diagram, classic, employed, markov, chains, allows, provides, way, estimate, parameter, easily, rightarrow, therefore, transitive, property, reiterate, definition, rearranged, assume, ages, year, instance, younger, easy, argument, multiplied, actually, feature, our, contract, notice, higher, vice, versa, expression, limitations, concerning, equals, implicates, said, sustained, without, infecting, grow, measure, transferable, quantity, determines, whether, remain, constant, become, throughout, decrease, always, straightforward, approximated, per, poisson, poissonian, erdős, rényi, moment, called, node, representation, edge, pair, nodes, represents, them, own, known, responsible, transmitting, locally, meaning, local, neighborhood, takes, form, written, excess, common, explanation, holds, doubling, analogous, tags, progresses, becomes, frenetic, run, previously, hunt, down, until, crashes, zero, peak, gradual, decline, seen, reality, grows, formally, belong, incorporate, features, into, sociality, opinion, wealth, profoundly, propagation, typically, partial, contrast, classical, described, following, derivation, provide, rigorous, starting, transition, expressed, derivatives, while, building, differentiable, respect, words, develop, dealing, assigned, representing, specific, stage, having, variable, estimating, probability, distributions, allowing, variation, depend, chance, variations, illness, small, determined, scrutiny, assort, mostly, subgroup, rarely, widespread, most, londoners, turkish, teenagers, two, examples, outside, standard, tractable, everybody, developed, countries, low, infant, much, good, makes, predictions, results, correct, recently, exchange, simpler, nonpharmaceutical, spite, complexity, requiring, high, power, criticized, simplifying, unrealistic, still, informing, decisions, regarding, mitigation, suppression, accurately, calibrated, 1920s, saw, emergence, 1928, both, relationship, successful, behavior, outbreaks, very, similar, recorded, reed, frost, early, 20th, century, william, explain, behaviour, law, ronald, earliest, carried, trained, physician, defend, inoculating, showed, work, preceded, understanding, first, scientist, systematically, tried, quantify, his, 1662, studied, listings, numbers, published, weekly, beginning, competing, risks, according, established, among, epidemiologists, political, observations, made, mechanisms, evaluate, strategies, project, progress, show, likely, plant, collected, along, various, calculate, programs, decide, avoid, trial, patterns, etc, understand, redirected, free, encyclopedia, item, projects, printable, version, download, print, export, switch, legacy, parser, get, shortened, url, information, permanent, what, here, actions, english, talk, українська, italiano, հայերեն, magyar, galego, suomi, español, deutsch, العربية, personal, pages, portal, learn, contribute, current, events, navigation, jump, content,

Text of the page (random words):
c 1 r_ 0 frac a l rightarrow r_ 0 frac l a this provides a simple way to estimate the parameter r 0 using easily available data for a population with an exponential age distribution r 0 1 l a 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 compartmental models in epidemiology edit main article compartmental models in epidemiology compartmental models are formulated as markov chains 25 a classic compartmental model in epidemiology is the sir model which may be used as a simple model for modelling epidemics multiple other types of compartmental models are also employed the sir model edit diagram of the sir model with initial values s 0 997 i 0 3 r 0 0 textstyle s 0 997 i 0 3 r 0 0 and rates for infection β 0 4 textstyle beta 0 4 and for recovery γ 0 04 textstyle gamma 0 04 animation of the sir model with initial values s 0 997 i 0 3 r 0 0 textstyle s 0 997 i 0 3 r 0 0 and rate of recovery γ 0 04 textstyle gamma 0 04 the animation shows the effect of reducing the rate of infection from β 0 5 textstyle beta 0 5 to β 0 12 textstyle beta 0 12 if there is no medicine or vaccination available it is only possible to reduce the infection rate often referred to as flattening the curve by appropriate measures such as social distancing in 1927 w o kermack and a g mckendrick created a model in which they considered a fixed population with only three compartments susceptible s t displaystyle s t infected i t displaystyle i t and recovered r t displaystyle r t the compartments used for this model consist of three classes 26 s t displaystyle s t or those susceptible to the disease of the population i t displaystyle i t denotes the individuals of the population who have been infected with the disease and are capable of spreading the disease to those in the susceptible category r t displaystyle r t is the compartment used for the individuals of the population 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 other compartmental models edit there are many 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 citation needed continuous models from seirs to density equation edit whereas compartmental models like seirs consider discrete states of a disease continuous models describe the evolution of the density of individuals in the semi infinite space of infection age the time passed since the last infection interestingly such an equation is derived from an ordinary differential equation for the interplay of viral load and immune response 27 as in the refractory density approach used for neurons in mathematical neuroscience infectious disease dynamics edit 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 28 research topics include antigenic shift epidemiological networks evolution and spread of resistance immuno epidemiology intra host dynamics pandemic pathogen population genetics persistence of pathogens within hosts phylodynamics role and identification of infection reservoirs role of host genetic factors spatial epidemiology statistical and mathematical tools and innovations strain biology structure and interactions transmission spread and control of infection virulence mathematics of mass vaccination edit 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 and its transmission dies out 29 thus a disease can be eliminated from a population if enough individuals are immune due to either vaccination or recovery from prior exposure to disease for example smallpox eradication with the last wild case in 1977 and certification of the eradication of indigenous transmission of 2 of the 3 types of wild poliovirus type 2 in 2015 after the last reported case in 1999 and type 3 in 2019 after the last reported case in 2012 30 the herd immunity level will be denoted q recall that for a stable state citation needed r 0 s 1 displaystyle r_ 0 cdot s 1 in turn r 0 n s μ n e t l μ n e min t l t s e t l e min t l t s displaystyle r_ 0 frac n s frac mu n operatorname e t_ l mu n operatorname e min t_ l t_ s frac operatorname e t_ l operatorname e min t_ l t_ s which is approximately citation needed e t l e t s 1 λ μ β n v displaystyle frac operatorname operatorname e t_ l operatorname operatorname e t_ s 1 frac lambda mu frac beta n v graph of herd immunity threshold vs basic reproduction number with selected diseases 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 r 0 1 q 1 1 q 1 r 0 q 1 1 r 0 displaystyle begin aligned r_ 0 cdot 1 q 1 6pt 1 q frac 1 r_ 0 6pt q 1 frac 1 r_ 0 end aligned remember that this is the threshold level die out of transmission will only occur if the proportion of immune individuals exceeds this level due to a mass vaccination programme we have just calculated the critical immunization threshold denoted q c it is the minimum proportion of the population that must be immunized at birth or close to birth in order for the infection to die out in the population q c 1 1 r 0 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 lim t s t e 0 λ t d t 1 p displaystyle lim _ t to infty s t e int _ 0 infty lambda t dt 1 p hence p 1 e 0 β i t d t 1 e r 0 p displaystyle p 1 e int _ 0 infty beta i t dt 1 e r_ 0 p solving for r 0 displaystyle r_ 0 we obtain r 0 ln 1 p p displaystyle r_ 0 frac ln 1 p p when mass vaccination cannot exceed the herd immunity edit if the vaccine used is insufficiently effective or the required coverage cannot be reached the program may fail to exceed q c such a program will protect vaccinated individuals from disease but may change the dynamics of transmission citation needed suppose that a proportion of the population q where q q c is immunised at birth against an infection with r 0 1 the vaccination programme changes r 0 to r q where r q r 0 1 q 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 r 0 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 r 0 a and l assuming that life expectancy has not changed now citation needed r q l a q displaystyle r_ q frac l a_ q a q l r q l r 0 1 q displaystyle a_ q frac l r_ q frac l r_ 0 1 q but r 0 l a so a q l l a 1 q a l l 1 q a 1 q displaystyle a_ q frac l l a 1 q frac al l 1 q frac a 1 q thus the vaccination program may raise the average age of infection and unvaccinated individuals will experience a reduced force of infection due to the presence of the vaccinated group for a disease that leads to greater clinical severity in older populations the unvaccinated proportion of the population may experience the disease relatively later in life than would occur in the absence of vaccine when mass vaccination exceeds the herd immunity edit if a vaccination program 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 if elimination occurs everywhere at the same time then this can lead to eradication citation needed 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 immunization threshold citation needed eradication elimination everywhere at the same time such that the infectious agent dies out for example smallpox and rinderpest citation needed reliability edit models have the advantage of examining multiple outcomes simultaneously rather than making a single forecast models have shown broad degrees of reliability in past pandemics such as sars sars cov 2 31 swine flu mers and ebola 32 see also edit basic reproduction number compartmental models in epidemiology contact tracing critical community size disease surveillance ecosystem model force of infection landscape epidemiology next generation matrix pandemic risk factor sexual network waifw matrix references edit 1 2 daley gani 1999 p page needed hethcote hw 2000 the mathematics of infectious diseases siam review 42 4 599 653 bibcode 2000siamr 42 599h doi 10 1137 s0036144500371907 blower s bernoulli d 2004 an attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it reviews in medical virology 14 5 275 88 doi 10 1002 rmv 443 pmid 15334536 foppa im 2017 d bernoulli a pioneer of epidemiologic modeling 1760 a historical introduction to mathematical modeling of infectious diseases pp 1 20 doi 10 1016 b978 0 12 802260 3 00001 8 isbn 978 0 12 802260 3 hamer 1929 p page needed ross 1910 p page needed 1 2 brauer castillo chavez 2012 p page needed eisinger d thulke hh april 2008 spatial pattern formation facilitates eradication of infectious diseases the journal of applied ecology 45 2 415 423 bibcode 2008japec 45 415e doi 10 1111 j 1365 2664 2007 01439 x pmc 2326892 pmid 18784795 adam d april 2020 special report the simulations driving the world s response to covid 19 nature 580 7803 316 318 bibcode 2020natur 580 316a doi 10 1038 d41586 020 01003 6 pmid 32242115 squazzoni f polhill jg edmonds b ahrweiler p antosz p scholz g et al 2020 computational models that matter during a global pandemic outbreak a call to action journal of artificial societies and social simulation 23 2 doi 10 18564 jasss 4298 hdl 10037 19057 sridhar d majumder ms april 2020 modelling the pandemic bmj 369 m1567 doi 10 1136 bmj m1567 pmid 32317328 maziarz m zach m october 2020 agent based modelling for sars cov 2 epidemic prediction and intervention assessment a methodological appraisal journal of evaluation in clinical practice 26 5 1352 1360 doi 10 1111 jep 13459 pmc 7461315 pmid 32820573 huppert a katriel g 2013 mathematical modelling and prediction in infectious disease epidemiology clinical microbiology and infection 19 11 999 1005 doi 10 1111 1469 0691 12308 pmid 24266045 tembine h 3 november 2020 covid 19 data driven mean field type game perspective games 11 4 51 doi 10 3390 g11040051 hdl 10419 257469 nakamura gm monteiro ac cardoso gc martinez as 20 january 2017 efficient method for comprehensive computation of agent level epidemic dissemination in networks scientific reports 7 1 40885 arxiv 1606 07825 bibcode 2017natsr 740885n doi 10 1038 srep40885 pmc 5247741 pmid 28106086 nakamura gm cardoso gc martinez as february 2020 improved susceptible infectious susceptible epidemic equations based on uncertainties and autocorrelation functions royal society open science 7 2 191504 bibcode 2020rsos 791504n doi 10 1098 rsos 191504 pmc 7062106 pmid 32257317 dietz k 1967 epidemics and rumours a survey journal of the royal statistical society series a general 130 4 505 528 doi 10 2307 2982521 jstor 2982521 albi g bertaglia g boscheri w dimarco g pareschi l toscani g et al 2022 kinetic modelling of epidemic dynamics social contacts control with uncertain data and multiscale spatial dynamics predicting pandemics in a globally connected world volume 1 modeling and simulation in science engineering and technology pp 43 108 doi 10 1007 978 3 030 96562 4_3 isbn 978 3 030 96561 7 maier bf brockmann d 2020 effective containment explains subexponential growth in recent confirmed covid 19 cases in china science 368 6492 742 746 bibcode 2020sci 368 742m doi 10 1126 science abb4557 pmc 7164388 pmid 32269067 barabási 2016 p page needed kenah e robins jm september 2007 second look at the spread of epidemics on networks physical review e 76 3 pt 2 036113 arxiv q bio 0610057 bibcode 2007phrve 76c6113k doi 10 1103 physreve 76 036113 pmc 2215389 pmid 17930312 pastor satorras r castellano c van mieghem p vespignani a 2015 08 31 epidemic processes in complex networks reviews of modern physics 87 3 925 979 arxiv 1408 2701 bibcode 2015rvmp 87 925p doi 10 1103 revmodphys 87 925 rizi ak faqeeh a badie modiri a kivelä m 20 april 2022 epidemic spreading and digital contact tracing effects of heterogeneous mixing and quarantine failures physical review e 105 4 044313 arxiv 2103 12634 bibcode 2022phrve 105d4313r doi 10 1103 physreve 105 044313 pmid 35590624 miller e 2003 global burden of disease the vaccine book pp 37 50 doi 10 1016 b978 012107258 2 50005 6 isbn 978 0 12 107258 2 cosma shalizi 15 november 2018 data over space and time lecture 21 compartment models pdf carnegie mellon university retrieved september 19 2020 kermack wo mckendrick ag 1991 contributions to the mathematical theory of epidemics i 1927 bulletin of mathematical biology 53 1 2 33 55 bibcode 1927rspsa 115 700k doi 10 1007 bf02464423 jstor 94815 pmid 2059741 chizhov a pujo menjouet l schwalger t sensi m 2025 09 01 a refractory density approach to a multi scale seirs epidemic model infectious disease modelling 10 3 787 801 doi 10 1016 j idm 2025 03 004 issn 2468 0427 pmc 11982974 pmid 40213022 brauer f 2017 mathematical epidemiology past present and future infectious disease modelling 2 2 113 127 doi 10 1016 j idm 2017 02 001 pmc 6001967 pmid 29928732 britton t ball f trapman p 2020 a mathematical model reveals the influence of population heterogeneity on herd immunity to sars cov 2 science 369 6505 846 849 bibcode 2020sci 369 846b doi 10 1126 science abc6810 pmc 7331793 pmid 32576668 pollard aj bijker em 2021 a guide to vaccinology from basic principles to new developments nature reviews immunology 21 2 83 100 doi 10 1038 s41577 020 00479 7 pmc 7754704 pmid 33353987 renz a widerspick l dräger a 30 december 2020 fba reveals guanylate kinase as a potential target for antiviral therapies against sars cov 2 bioinformatics 36 supplement_2 i813 i821 doi 10 1093 bioinformatics btaa813 pmc 7773487 pmid 33381848 costris vas c schwartz ej smith rj november 2020 predicting covid 19 using past pandemics as a guide how reliable were mathematical models then and how reliab...