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| In order to [[mathematical modelling in epidemiology|model the progress of an epidemic]] in a large [[population]], the population diversity must be reduced to a few key characteristics which are relevant to the infection under consideration. For example, for most common childhood diseases that confer long-lasting immunity, it makes sense to divide the population into those who are [[susceptible]] to the disease, those who are [[infectious disease|infected]] and those who have recovered and are [[immune system|immune]]. These subdivisions of the population are called '''compartments'''.
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| ==The SIR model==
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| The '''SIR model''' labels these three compartments '''S''' = number susceptible, '''I''' =number infectious, and '''R''' =number recovered (immune). This is a good and simple model for many infectious diseases including [[measles]], [[mumps]] and [[rubella]]. | |
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| The letters also represent the number of people in each compartment at a particular time. To indicate that the numbers might vary over time (even if the total population size remains constant), we make the precise numbers a function of ''t'' (time): S(''t''), I(''t'') and R(''t''). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.
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| ===The SIR model is dynamic in three senses===
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| As implied by the variable function of ''t'', the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an [[endemic (epidemiology)|endemic]] disease with a short infectious period, such as [[measles]] in the UK prior to the introduction of a [[vaccination|vaccine]] in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(''t'')) over time. During an [[epidemic]], the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up as a result of offspring being born into the susceptible compartment.
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| [[File:Sirsys-p9.png|thumb| Blue=Susceptible, Green=Infected, and Red=Recovered]]
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| Each member of the population typically progresses from susceptible to infectious to removed. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments.
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| ===Transition rates===
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| For the full specification of the model, the arrows should be labeled with the transition rates between compartments.Between S and I, the transition rate is β I, where β is the contact rate, which takes into account the probability of getting the disease in a contact between a susceptible and an infectious subject.
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| Between I and R, the transition rate is ν (simply the rate of recovery or death). If the duration of the infection is denoted ''D'', then ν = 1/''D'', since an individual experiences one recovery in ''D'' units of time.
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| It is assumed that the permanence of each single subject in the epidemic states is a random variable with exponential distribution. More complex and realistic distributions (such as [[Erlang distribution]]) can be equally used with few modifications.
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| ==Bio-mathematical deterministic treatment of the SIR model==
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| ===The SIR model without vital dynamics===
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| Because a single epidemic outbreak moves faster than the normal birth-death rates in a given population, normal birth-death rates are excluded. The SIR system described above can be expressed by the following set of ordinary [[differential equations]]:
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| :<math> \frac{dS}{dt} = - \beta I S </math>
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| :<math> \frac{dI}{dt} = \beta I S - \nu I </math>
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| :<math> \frac{dR}{dt} = \nu I </math>
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| This model was for the first time proposed by O. Kermack and [[Anderson Gray McKendrick]], who had worked with the Nobel Laureate [[Ronald Ross]].
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| This system is [[non-linear]], and does not admit a generic [[analytic solution]]. Nevertheless, significant results can be derived analytically.
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| Firstly note that from:
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| :<math> \frac{dS}{dt} + \frac{dI}{dt} + \frac{dR}{dt} = 0 </math>
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| it follows that:
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| :<math> S(t) + I(t) + R(t) = \textrm{Constant} = N </math>
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| expressing in mathematical terms the constancy of population <math> N </math>. Note that the above relationship implies that one need only study the equation for two of the three variables.
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| Secondly, we note that the dynamics of the infectious class depends on the following ratio:
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| :<math> R_0 = N\frac{\beta}{\nu} </math>
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| the so-called [[basic reproduction number]] (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible
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| <ref name=Bailey1975>{{cite book |author=Bailey, Norman T. J. |title=The mathematical theory of infectious diseases and its applications |publisher=Griffin |location=London |year=1975 |isbn=0-85264-231-8 |edition=2nd}}</ref>
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| .<ref name=nunn2006>{{cite book |author=Sonia Altizer; Nunn, Charles |title=Infectious diseases in primates: behavior, ecology and evolution |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2006 |isbn=0-19-856585-2 |series=Oxford Series in Ecology and Evolution}}</ref> This idea can probably be more readily seen if we say that the typical time between contacts is <math> T_{c} = \beta^{-1}</math>, and the typical time until recovery is <math> T_{r} = \nu^{-1}</math>. From here it follows that, on average, the number of contacts by an infected individual with others ''before'' the infected has recovered is: <math> T_{r}/T_{c}. </math>
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| By dividing the first differential equation by the third, [[Separation of variables|separating the variables]] and integrating we get
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| :<math> S(t) = S(0) e^{-R_0(R(t) - R(0))} </math>
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| (where S(0) and R(0) are the initial numbers of, respectively, susceptible and removed subjects). Thus, in the limit <math>t \rightarrow +\infty</math>, the proportion of recovered individuals obeys the [[transcendental equation]]
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| :<math> R_{\infty} = 1 - S(0)e^{-R_0(R_{\infty} - R(0))} </math>
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| This equation shows that at the end of an epidemic, unless S(0)=0, not all individuals of the population have recovered, so some must remain susceptible. This means that the end of an epidemic is caused by the decline in the number of infected individuals rather than an absolute lack of susceptible subjects.
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| The role of the [[basic reproduction number]] is extremely important. In fact, upon rewriting the equation for infectious individuals as follows:
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| :<math> \frac{dI}{dt} = (R_0 S/N - 1) \nu I </math>
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| it yields that if:
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| :<math> R_{0} > \frac{N}{S(0)} ,</math>
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| then:
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| :<math> \frac{dI}{dt}(0) >0 ,</math>
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| i.e. there will be a proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if
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| :<math> R_{0} < \frac{N}{S(0)} ,</math>
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| then
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| :<math> \frac{dI}{dt}(0) <0 ,</math>
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| i.e. independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that the [[basic reproduction number]] is extremely important.
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| ===The force of infection===
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| Note that in the above model the function:
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| :<math> F = \beta I ,</math>
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| models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population <math>N</math>):
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| :<math> F = \beta \frac{I}{N} .</math>
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| Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process.
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| ===The SIR model with vital dynamics and constant population===
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| Considering a population characterized by a death rate <math>\mu</math> and birth rate equal to the death rate, and where a communicable disease is spreading. The model is:
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| :<math> \frac{dS}{dt} = \mu N - \mu S - \beta \frac{I}{N} S </math>
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| :<math> \frac{dI}{dt} = \beta \frac{I}{N} S - (\nu +\mu ) I </math>
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| :<math> \frac{dR}{dt} = \nu I - \mu R </math>
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| for which it holds:
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| :<math>S+I+R=N.</math>
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| Also in this case we may define a [[basic reproduction number]] :
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| :<math> R_0 = \frac{ \beta }{\mu+\nu}, </math>
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| which has threshold properties. In fact, independently from biologically meaningful initial values:
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| :<math> \left(S(0),I(0),R(0)\right) \in \left\{(S,I,R)\in [0,N]^3 : S+I+R = N \right\} </math>
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| one can show that:
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| :<math> R_0 \le 1 \Rightarrow \lim_{t \rightarrow +\infty} \left(S(t),I(t),R(t)\right) = \textrm{DFE} = \left(N,0,0\right) </math>
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| :<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \rightarrow +\infty} \left(S(t),I(t),R(t)\right) = \textrm{EE} = \left(\frac{N}{R_0},\frac{\mu}{\beta}\left(R_0-1\right)N,\frac{\nu}{\beta} \left(R_0-1\right)N\right). </math>
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| The point DFE is called the disease free equilibrium, whereas the point EE is called the Endemic Equilibrium. Since, with heuristic arguments, one may show that <math>R_{0}</math> may be read as the average number of infections caused by a single infectious subject in a wholly susceptible population, the above relationship biologically means that if this number is less or equal than one the disease get extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.
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| ===Variable contact rates and pluriannual or chaotic epidemics===
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| It is well known that the probability of getting a disease is not constant in the time. Some diseases are seasonal, such as the [[Coronavirus|common cold viruses]], which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during the school holidays the probability of getting such a disease dramatically decreases.
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| As a consequence, for many classes of diseases one should consider a force of infection with periodically ('seasonal') varying contact rate
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| :<math> F = \beta(t) \frac{I}{N} , \beta(t+T)=\beta(t)</math>
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| with period T equal to one year.
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| Thus, our model becomes
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| :<math> \frac{dS}{dt} = \mu N - \mu S - \beta(t) \frac{I}{N} S </math>
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| :<math> \frac{dI}{dt} = \beta(t) \frac{I}{N} S - (\nu +\mu ) I </math>
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| (the dynamics of recovered easily follows from <math>R=N-S-I</math>), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if:
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| :<math> \frac{1}{T} \int_{0}^{T}\frac{\beta(t)}{\mu+\nu}dt < 1 \Rightarrow \lim_{t \rightarrow +\infty} \left(S(t),I(t)\right) = DFE = \left(N,0\right), </math>
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| whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.
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| This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near the endemic equilibrium.
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| Remarkably, in some cases the behavior may also be quasi-periodic or even chaotic.
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| ==Modelling mass vaccination programmes==
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| ===Vaccinating the newborns===
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| In presence of a communicable diseases, one of main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Let us consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate <math>P \in (0,1)</math>:
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| :<math> \frac{dS}{dt} = \mu N (1-P) - \mu S - \beta \frac{I}{N} S </math>
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| :<math> \frac{dI}{dt} = \beta \frac{I}{N} S - (\mu+\nu) I </math>
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| :<math> \frac{dV}{dt} = \mu N P - \mu V</math>
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| where V is the class of vaccinated subjects. It is immediate to show that:
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| :<math> \lim_{t \rightarrow +\infty} V(t)= N P,</math>
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| thus we shall deal with the long term behavior of S and I, for which it holds that:
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| :<math> R_0 (1-P) \le 1 \Rightarrow \lim_{t \rightarrow +\infty} \left(S(t),I(t)\right) = DFE = \left(N \left(1-P\right),0\right) </math>
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| :<math> R_0 (1-P) > 1 , I(0)> 0 \Rightarrow \lim_{t \rightarrow +\infty} \left(S(t),I(t)\right) = EE = \left(\frac{N}{R_0(1-P)},N \left(R_0 (1-P)-1\right)\right). </math>
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| In other words if
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| :<math> P \ge P^{*}= 1-\frac{1}{R_0} </math>
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| the vaccination program is successful in eradicating the disease, on the contrary it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that for a disease whose [[basic reproduction number]] may be as high as 18 one should have to vaccinate 94.4% of newborns in order to eradicate the disease.
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| ===Vaccination and information===
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| Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects:
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| :<math> P=P(I),P^{\prime}(I)>0.</math>
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| In such a case the eradication condition becomes:
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| :<math> P(0) \ge P^{*},</math>
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| i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.
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| ===Vaccination of non newborns===
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| In case there also are vaccinations of non newborn at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:
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| :<math> \frac{dS}{dt} = \mu N (1-P) - \mu S - \rho S - \beta \frac{I}{N} S </math>
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| :<math> \frac{dV}{dt} = \mu N P + \rho S - \mu V</math>
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| leading to the following eradication condition:
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| :<math> P \ge 1- \left(1+\frac{\rho}{\mu}\right)\frac{1}{R_0} </math>
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| ===Pulse vaccination strategy===
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| This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects:
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| :<math> \frac{dS}{dt} = \mu N - \mu S - \beta \frac{I}{N} S, S(n T^+) = (1-p) S(n T^-) n=0,1,2,\dots </math>
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| :<math> \frac{dV}{dt} = - \mu V, V(n T^+) = V(n T^-) + p S(n T^-) n=0,1,2,\dots</math>
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| It is easy to see that by setting {{math|1=''I'' = 0}} one obtains that the dynamics of the susceptible subjects is given by:
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| :<math> S^*(t) = 1- \frac{p}{1-(1-p)E^{-\mu T}}E^{-\mu MOD(t,T)} </math>
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| and that the eradication condition is:
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| :<math> R_0 \int_{0}^{T}{S^*(t)dt} < 1 </math>
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| ==The SEIR model==
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| For many important infections there is a significant incubation period during which the individual has been infected but is not yet infectious themselves. During this period the individual is in compartment E (for exposed).
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| [[File:SEIR.PNG|800px|center|SEIR compartmental model]]
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| Assuming that the incubation period is a random variable with exponential distribution with
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| parameter a (i.e. the average incubation period is <math>a^{-1}</math>), and also assuming the presence of vital dynamics with birth rate equal to death rate, we have the model:
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| :<math> \frac{dS}{dt} = \mu N - \mu S - \beta \frac{I}{N} S </math>
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| :<math> \frac{dE}{dt} = \beta \frac{I}{N} S - (\mu +a ) E </math>
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| :<math> \frac{dI}{dt} = a E - (\nu +\mu ) I </math>
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| :<math> \frac{dR}{dt} = \nu I - \mu R. </math>
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| Of course, we have that <math>S+E+I+R=N</math>.
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| For this model, the basic reproduction number is:
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| <math>R_0 = \frac{a}{\mu+a}\frac{\beta}{\mu+\nu}.</math>
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| Similarly to the SIR model, also in this case we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently form biologically meaningful initial conditions
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| :<math> \left(S(0),E(0),I(0),R(0)\right) \in \left\{(S,E,I,R)\in [0,N]^4 : S \ge 0, E \ge 0, I\ge 0, R\ge 0, S+E+I+R = N \right\} </math>
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| it holds that:
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| :<math> R_0 \le 1 \Rightarrow \lim_{t \rightarrow +\infty} \left(S(t),E(t),I(t),R(t)\right) = DFE = \left(N,0,0,0\right) </math>
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| :<math> R_0 > 1 , I(0)> 0 \Rightarrow \lim_{t \rightarrow +\infty} \left(S(t),E(t),I(t),R(t)\right) = EE. </math>
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| In case of periodically varying contact rate <math>\beta(t)</math> the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:
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| :<math> \frac{dE_1}{dt} = \beta(t) I_1 - (\nu +a ) E_1 </math>
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| :<math> \frac{dI_1}{dt} = a E_1 - (\nu +\mu ) I_1 </math>
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| is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).
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| ==Elaborations on the basic SIR model==
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| ===The MSIR model===
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| For many infections, including [[measles]], babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the [[placenta]] and additionally through [[colostrum]]). This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.
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| [[File:MSIR.PNG|800px|center|MSIR compartmental model]]
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| ===Carrier state===
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| Some people who have had an infectious disease such as [[tuberculosis]] never completely recover and continue to [[asymptomatic carrier|carry]] the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably [[Mary Mallon]], who infected 22 people with [[typhoid fever]]. The carrier compartment is labelled C.
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| [[File:SIR model with carrier.PNG|500px|center|SIR compartmental model with carrier class, C]]
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| ==The SIS model==
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| [[File:Sissys.png|thumb|Susceptibles and infected get equilibrated.]]
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| Some infections, for example those from the [[common cold]] and [[influenza]], do not confer any long lasting immunity. Such infections do not give immunization upon recovery from infection, and individuals become susceptible again.
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| [[File:SIS.PNG|400px|center|SIS compartmental model]]
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| We have the model:
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| :<math> \frac{dS}{dt} = - \beta S I + \gamma I </math>
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| :<math> \frac{dI}{dt} = \beta S I - \gamma I </math>
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| Note that denoting with N the total population it holds that:
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| <math>\frac{dS}{dt} + \frac{dI}{dt} = 0 \Rightarrow S(t)+I(t) = N</math>
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| it follows that:
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| :<math> \frac{dI}{dt} = \left( \beta N - \gamma\right) I - \beta I^2 </math>
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| i.e. the dynamics of infectious is ruled by a logistic equation, so that <math>\forall I(0) > 0</math>:
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| :<math> \frac{\beta N}{\gamma} \le 1 \Rightarrow \lim_{t \rightarrow +\infty}I(t)=0 </math>
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| :<math> \frac{\beta N}{\gamma} > 1 \Rightarrow \lim_{t \rightarrow +\infty}I(t)=\frac{\beta N - \gamma}{\beta} </math>
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| ==The influence of age: age-structured models==
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| Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic classes <math>s(t,a),i(t,a),r(t,a)</math> (to limit ourselves to the susceptible-infectious-removed scheme) such that:
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| :<math>S(t)=\int_0^{a_M}{s(t,a)da}</math>
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| :<math>I(t)=\int_0^{a_M}{i(t,a)da}</math>
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| :<math>R(t)=\int_0^{a_M}{r(t,a)da}</math>
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| (where <math>a_M \le +\infty</math> is the maximum admissible age)and their dynamics is not described, as one might think, by "simple" partial differential equations, but by [[integro-differential equation]]s:
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| :<math>\partial_t s(t,a) + \partial_a s(t,a) = -\mu(a) s(a,t) - s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} </math>
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| :<math>\partial_t i(t,a) + \partial_a i(t,a) = s(a,t)\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\mu(a) i(a,t) - \nu(a)i(a,t) </math>
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| :<math>\partial_t r(t,a) + \partial_a r(t,a) = -\mu(a) r(a,t) + \nu(a)i(a,t) </math>
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| where:
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| :<math>F(a,t,i(\cdot,\cdot))=\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1}</math>
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| is the force of infection, which, of course, will depend, though the contact kernel <math> k(a,a_1;t) </math> on the interactions between the ages.
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| Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:
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| :<math>i(t,0)=r(t,0)=0</math>
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| but that are nonlocal for the density of susceptible newborns:
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| :<math>s(t,0)= \int_{0}^{a_M}{\left(\varphi_s(a) s(a,t)+\varphi_i(a) i(a,t)+\varphi_r(a) r(a,t)\right) da} </math>
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| where <math>\varphi_j(a), j=s,i,r</math> are the fertilities of the adults.
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| Moreover, defining now the density of the total population <math>n(t,a)=s(t,a)+i(t,a)+r(t,a)</math> one obtains:
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| :<math>\partial_t n(t,a) + \partial_a n(t,a) = -\mu(a) n(a,t) </math>
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| In the simplest case of equal fertilities in the three epidemic classes, we have that in oder to have demographic equilibrium the following necessary and sufficient condition linking the fertility <math>\varphi(.)</math> with the mortality <math>\mu(a)</math> must hold:
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| :<math> 1 = \int_0^{a_M}{\varphi(a) \exp\left(- \int_0^a{\mu(q)dq} \right)da } </math>
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| and the demographic equilibrium is
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| :<math>n^*(a)=C \exp\left(- \int_0^a{\mu(q)dq}\right),</math>
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| automatically ensuring the existence of the disease-free solution:
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| :<math>DFS(a)= \left(n^*(a),0,0\right).</math>
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| A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.
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| ==Deterministic versus stochastic epidemic models==
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| It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations. Moreover, even in case of large populations, as pointed out among the firsts by [[M. S. Bartlett]], in some cases deterministic models should cautiously be used. For example in case of seasonally varying contact rates the number of infectious subjects may reduce to infinitesimal values, thus maybe invalidating some results that are obtained in the field of chaotic epidemics.
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| ==The GEMF model: Generalized Epidemic Mean-Field model==
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| The generalized epidemic mean-field model ('''GEMF''') is a novel and generalized formulation of the epidemic spreading problem and a modeling solution. This modeling approach considers a spreading process among a group of nodes that can be in different compartments (susceptible, infected, recovered, and vaccinated), and where the nodes are connected through a multi-layer network. It is suitable for a large class of individual-based spreading scenarios. GEMF allows for simple and rapid development of software to simulate multi-compartment and multi-layer epidemic models.<ref>Darabi Sahneh F., Scoglio C., Van Mieghem P., "Generalized Epidemic Mean-Field Model for Spreading Processes over Multi-Layer Complex networks," IEEE/ACM Transactions on Networking, vol. 21, no.5, 1609-1620, 2013.</ref>
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| ==See also==
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| *[[Mathematical modelling in epidemiology]]
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| *[[Modifiable Areal Unit Problem]]
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| *[[Next-generation matrix]]
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| *[[Risk assessment]]
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| *[[Attack rate]]
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| ==References==
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| {{reflist}}
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| ==Bibliography==
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| *{{cite book |author=May, Robert M.; Anderson, Roy M. |title=Infectious diseases of humans: dynamics and control |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=1991 |pages= |isbn=0-19-854040-X }}
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| *V. Capasso, The Mathematical Structure of Epidemic Systems, Springer Verlag (1993)
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| *{{cite journal |author=McKendrick AG |title=Applications of mathematics to medical problems |journal=Proceedings of the Edinburgh Mathematical Society |volume=44 |pages=98–130 |year=1925 |doi=10.1017/S0013091500034428 }}<br/>Reprinted with commentary in {{cite book |author=Johnson, Norman L.; Kotz, Samuel |title=Breakthroughs in statistics |publisher=Springer-Verlag |location=Berlin |year=1992 |pages= |isbn=0-387-94989-5 |volume=3}}
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| *{{cite journal |author=Kermack WO, McKendrick AG |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society A |volume=115 |issue=772 |pages=700–721 |date=August 1, 1927 |doi=10.1098/rspa.1927.0118}}
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| *{{cite journal |author=Kermack WO, McKendrick AG |title=Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity |journal=Proceedings of the Royal Society A |volume=138 |issue=834 |pages=55–83 |date=October 1, 1932 |doi=10.1098/rspa.1932.0171}}
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| *{{cite journal |author=Kermack WO, McKendrick AG |title=Contributions to the Mathematical Theory of Epidemics. III. Further Studies of the Problem of Endemicity |journal=Proceedings of the Royal Society A |volume=141 |issue=843 |pages=94–122 |date=July 3, 1933 |doi=10.1098/rspa.1933.0106}}
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| *{{cite journal |author=Bartlett MS |title=Measles periodicity and community size |journal=Journal of the Royal Statistical Society, Series A |volume=120 |pages=48–70 |year=1957 |doi=10.2307/2342553 |issue=1 |jstor=2342553}}
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| *{{cite journal |author=Hethcote H |title=Qualitative analyses of communicable disease models |journal=Math. Biosci. |volume=28 |issue= 3-4|pages=335–356 |year=1976 |url=http://www.math.uiowa.edu/~hethcote/PDFs/1976MathBiosci.pdf |doi=10.1016/0025-5564(76)90132-2}}
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| *{{cite journal |author=Olsen LF, Schaffer WM |title=Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics |journal=Science |volume=249 |issue=4968 |pages=499–504 |date=August 1990 |pmid=2382131 |url=http://www.sciencemag.org/cgi/pmidlookup?view=long&pmid=2382131 |doi=10.1126/science.2382131}}
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| *{{cite journal |author=Inaba H |title=Threshold and stability results for an age-structured epidemic model |journal=J Math Biol |volume=28 |issue=4 |pages=411–34 |year=1990 |pmid=2384720 }}
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| *{{cite journal |author=Agur Z, Cojocaru L, Mazor G, Anderson RM, Danon YL |title=Pulse mass measles vaccination across age cohorts |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=90 |issue=24 |pages=11698–702 |date=December 1993 |pmid=8265611 |pmc=48051 |doi=10.1073/pnas.90.24.11698 }}
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| *{{cite journal |author=Kuznetsov YA, Piccardi C |title=Bifurcation analysis pf periodic SEIR and SIR epidemic models |journal=J Math Biol |volume=32 |issue=2 |pages=109–21 |year=1994 |pmid=8145028 |doi=10.1007/BF00163027 }}
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| *{{cite journal |author=Li MY, Muldowney JS |title=Global stability for the SEIR model in epidemiology |journal=Math Biosci |volume=125 |issue=2 |pages=155–64 |date=February 1995 |pmid=7881192 |url=http://linkinghub.elsevier.com/retrieve/pii/0025556495927565 |doi=10.1016/0025-5564(95)92756-5}}
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| *{{cite book |author=Nasell, I. |chapter=Measles outbreaks are not chaotic |editor=Blower, Sally; Castillo-Chávez, Carlos |title=Mathematical approaches for emerging and reemerging infectious diseases: an introduction |publisher=Springer |location=Berlin |year=2002 |pages=85–115 |isbn=0-387-95354-X }}
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| *{{cite journal |author=d'Onofrio A |title=Stability properties of pulse vaccination strategy in SEIR epidemic model |journal=Math Biosci |volume=179 |issue=1 |pages=57–72 |year=2002 |pmid=12047921 |doi=10.1016/S0025-5564(02)00095-0 }}
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| *{{cite journal |author=d'Onofrio A |title=Mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times |journal=Applied Mathematics and Computation |volume=151 |issue=1 |pages=181–7 |year=2004 |doi=10.1016/S0096-3003(03)00331-X}}
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| *{{cite journal |author=Reluga TC, Bauch CT, Galvani AP |title=Evolving public perceptions and stability in vaccine uptake |journal=Math Biosci |volume=204 |issue=2 |pages=185–98 |date=December 2006 |pmid=17056073 |doi=10.1016/j.mbs.2006.08.015 |url=http://linkinghub.elsevier.com/retrieve/pii/S0025-5564(06)00150-7}}
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| *{{cite journal |author=d'Onofrio A, Manfredi P, Salinelli E |title=Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases |journal=Th. Pop. Biol. |pages=301–17 |year=2007}}
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| *{{cite journal |author=Schaffer WM, Bronnikova TV |title=Parametric dependence in model epidemics |journal=J. Biol. Dynam. |volume=1 |issue= 2|pages=183–195 |year=2007 |url=http://bill.srnr.arizona.edu/mss/UPD-1%20Published.pdf |doi=10.1080/17513750601174216}}
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| *{{cite journal |author=Stone L, Olinky R, Huppert A |title=Seasonal dynamics of recurrent epidemics |journal=Nature |volume=446 |issue=7135 |pages=533–6 |date=March 2007 |pmid=17392785 |doi=10.1038/nature05638}}
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| *{{cite book |author=Vynnycky, E., White, R.G. |editor=Vynnycky, E., White, R.G. |title=An Introduction to Infectious Disease Modelling |publisher=Oxford University Press |location=Oxford |year=2010 |pages=368 |isbn=0-19-856576-3 |url=http://anintroductiontoinfectiousdiseasemodelling.com }}
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| ==External links==
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| * the [[Spatiotemporal Epidemiological Modeler]] (STEM) is an open source Epidemiological modeling system that uses the [[Eclipse Modeling Framework]] from the [[Eclipse Foundation]] to allow rapid development of new models for infectious disease. External [http://www.eclipse.org/stem/ STEM] site at [http://www.eclipse.org/ Eclipse].
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| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Epidemiology:_The_SIR_model SIR model: Online experiments with JSXGraph]
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| * [http://epl.se/simulation/?p=1 SEIR Model simulator] Free to use epidemics simulator using SEIR model. Systems Dynamics implementation of the SEIR model.
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| {{Computer modeling|state=expanded}}
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| {{DEFAULTSORT:Compartmental Models In Epidemiology}}
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| [[Category:Epidemiology]]
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| [[Category:Scientific modeling]]
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