Los modelos compartimentales simplifican el modelado matemático de enfermedades infecciosas . La población se asigna a los compartimentos con etiquetas - por ejemplo, S , I , o R , ( S usceptible, I nfectious, o R ecovered). Las personas pueden progresar entre compartimentos. El orden de las etiquetas suele mostrar los patrones de flujo entre los compartimentos; por ejemplo SEIS significa susceptible, expuesto, infeccioso y luego susceptible de nuevo.
El origen de estos modelos se remonta a principios del siglo XX, siendo obras importantes las de Ross [1] en 1916, Ross y Hudson en 1917, [2] [3] Kermack y McKendrick en 1927 [4] y Kendall en 1956 [5 ]
Los modelos se ejecutan con mayor frecuencia con ecuaciones diferenciales ordinarias (que son deterministas), pero también se pueden usar con un marco estocástico (aleatorio), que es más realista pero mucho más complicado de analizar.
Los modelos intentan predecir cosas como cómo se propaga una enfermedad, o el número total de infectados, o la duración de una epidemia, y estimar varios parámetros epidemiológicos como el número reproductivo . Dichos modelos pueden mostrar cómo las diferentes intervenciones de salud pública pueden afectar el resultado de la epidemia, por ejemplo, cuál es la técnica más eficiente para emitir un número limitado de vacunas en una población determinada.
El modelo SIR
El modelo SIR [6] [7] [8] [9] es uno de los modelos compartimentales más simples, y muchos modelos son derivados de esta forma básica. El modelo consta de tres compartimentos: -
- S : El número de s individuos usceptible. Cuando un individuo susceptible y uno infeccioso entran en "contacto infeccioso", el individuo susceptible contrae la enfermedad y pasa al compartimento infeccioso.
- I : El número de i individuos nfectious. Estos son individuos que han sido infectados y son capaces de infectar a individuos susceptibles.
- R para el número de r emoved (e inmune) o individuos fallecidos. Se trata de personas que se han infectado y se han recuperado de la enfermedad y han entrado en el compartimento extraído o han muerto. Se asume que el número de muertes es insignificante con respecto a la población total. Este compartimento también puede ser llamado " r ecovered" o " r esistant".
Este modelo es razonablemente predictivo [10] para enfermedades infecciosas que se transmiten de persona a persona y en las que la recuperación confiere una resistencia duradera, como el sarampión , las paperas y la rubéola .
Estas variables ( S , I y R ) representan el número de personas en cada compartimento en un momento determinado. Para representar que el número de individuos susceptibles, infecciosos y removidos puede variar con el tiempo (incluso si el tamaño total de la población permanece constante), hacemos que los números precisos sean una función de t (tiempo): S ( t ), I ( t ) y R ( t ). Para una enfermedad específica en una población específica, estas funciones pueden resolverse con el fin de predecir posibles brotes y controlarlos. [10]
Como lo implica la función variable de t , el modelo es dinámico en el sentido de que los números en cada compartimento pueden fluctuar con el tiempo. La importancia de este aspecto dinámico es más obvia en una enfermedad endémica con un período infeccioso corto, como el sarampión en el Reino Unido antes de la introducción de una vacuna en 1968. Estas enfermedades tienden a ocurrir en ciclos de brotes debido a la variación en el número de susceptibles (S ( t )) a lo largo del tiempo. Durante una epidemia , el número de individuos susceptibles disminuye rápidamente a medida que más de ellos se infectan y, por lo tanto, ingresan a los compartimentos infecciosos y eliminados. La enfermedad no puede volver a brotar hasta que se haya recuperado el número de susceptibles, por ejemplo, como resultado del nacimiento de una descendencia en el compartimento susceptible.
Cada miembro de la población progresa típicamente de susceptible a infeccioso para recuperarse. Esto se puede mostrar como un diagrama de flujo en el que las cajas representan los diferentes compartimentos y las flechas la transición entre compartimentos, es decir
Tasas de transición
Para la especificación completa del modelo, las flechas deben estar etiquetadas con las tasas de transición entre compartimentos. Entre S e I , se supone que la tasa de transición es d (S / N) / dt = -βSI / N 2 , donde N es la población total, β es el número promedio de contactos por persona por tiempo, multiplicado por la probabilidad de transmisión de la enfermedad en un contacto entre un sujeto susceptible y uno infeccioso, y SI / N 2 es la fracción de esos contactos entre un individuo infeccioso y susceptible que dan como resultado que la persona susceptible se infecte. (Esto es matemáticamente similar a la ley de acción de masas en química en la que las colisiones aleatorias entre moléculas dan como resultado una reacción química y la tasa fraccional es proporcional a la concentración de los dos reactivos).
Entre I y R , la tasa de transición se supone que es proporcional al número de individuos infecciosa que se γ I . Esto equivale a suponer que la probabilidad de que un individuo infeccioso se recupere en cualquier intervalo de tiempo dt es simplemente γ dt . Si un individuo es infeccioso para un promedio período de tiempo D , entonces γ = 1 / D . Esto también es equivalente a la suposición de que el tiempo que pasa un individuo en estado infeccioso es una variable aleatoria con una distribución exponencial . El modelo SIR "clásico" puede modificarse utilizando distribuciones más complejas y realistas para la tasa de transición de IR (por ejemplo, la distribución de Erlang [11] ).
Para el caso especial en el que no hay remoción del compartimento infeccioso (γ = 0), el modelo SIR se reduce a un modelo SI muy simple, que tiene una solución logística , en el que cada individuo eventualmente se infecta.
El modelo SIR sin dinámica vital
La dinámica de una epidemia, por ejemplo, la gripe , a menudo es mucho más rápida que la dinámica del nacimiento y la muerte, por lo tanto, el nacimiento y la muerte a menudo se omiten en modelos compartimentales simples. El sistema SIR sin la llamada dinámica vital (nacimiento y muerte, a veces llamada demografía) descrito anteriormente se puede expresar mediante el siguiente conjunto de ecuaciones diferenciales ordinarias : [7] [12]
dónde es el stock de población susceptible, es el stock de infectados, es el stock de población eliminada (ya sea por muerte o recuperación), y es la suma de estos tres.
Este modelo fue propuesto por primera vez por William Ogilvy Kermack y Anderson Gray McKendrick como un caso especial de lo que ahora llamamos teoría de Kermack-McKendrick , y siguió al trabajo que McKendrick había hecho con Ronald Ross .
Este sistema no es lineal , sin embargo, es posible derivar su solución analítica en forma implícita. [6] En primer lugar, tenga en cuenta que de:
resulta que:
expresando en términos matemáticos la constancia de la población . Tenga en cuenta que la relación anterior implica que solo es necesario estudiar la ecuación para dos de las tres variables.
En segundo lugar, observamos que la dinámica de la clase infecciosa depende de la siguiente proporción:
el llamado número de reproducción básico (también llamado índice de reproducción básico). Esta relación se deriva como el número esperado de nuevas infecciones (estas nuevas infecciones a veces se denominan infecciones secundarias) de una sola infección en una población en la que todos los sujetos son susceptibles. [13] [14] Esta idea probablemente se pueda ver más fácilmente si decimos que el tiempo típico entre contactos es, y el tiempo típico hasta la extracción es . De aquí se deduce que, en promedio, el número de contactos de un individuo infeccioso con otros antes de que se haya eliminado el infeccioso es:
Dividiendo la primera ecuación diferencial por la tercera, separando las variables e integrando obtenemos
dónde y son los números iniciales de sujetos susceptibles y removidos, respectivamente. Escritura para la proporción inicial de individuos susceptibles, y y para la proporción de individuos susceptibles y removidos respectivamente en el límite uno tiene
(tenga en cuenta que el compartimento infeccioso se vacía en este límite). Esta ecuación trascendental tiene una solución en términos de la función W de Lambert , [15] a saber
Esto muestra que al final de una epidemia que se ajusta a los supuestos simples del modelo SIR, a menos que , no se han eliminado todos los individuos de la población, por lo que algunos deben seguir siendo susceptibles. Una fuerza impulsora que lleva al final de una epidemia es la disminución del número de individuos infecciosos. Por lo general, la epidemia no termina debido a una falta total de individuos susceptibles.
El papel tanto del número de reproducción básico como de la susceptibilidad inicial es extremadamente importante. De hecho, al reescribir la ecuación para individuos infecciosos de la siguiente manera:
da que si:
luego:
es decir, habrá un brote epidémico adecuado con un aumento del número de infecciosos (que puede llegar a una fracción considerable de la población). Por el contrario, si
luego
es decir, independientemente del tamaño inicial de la población susceptible, la enfermedad nunca puede causar un brote epidémico adecuado. Como consecuencia, está claro que tanto el número de reproducción básico como la susceptibilidad inicial son extremadamente importantes.
La fuerza de la infección
Tenga en cuenta que en el modelo anterior la función:
modela la tasa de transición del compartimento de individuos susceptibles al compartimento de individuos infecciosos, por lo que se denomina fuerza de infección . Sin embargo, para grandes clases de enfermedades transmisibles es más realista considerar una fuerza de infección que no depende del número absoluto de sujetos infecciosos, sino de su fracción (con respecto a la población total constante):
Capasso [16] y, posteriormente, otros autores han propuesto fuerzas de infección no lineales para modelar de forma más realista el proceso de contagio.
Soluciones analíticas exactas al modelo SIR
En 2014, Harko y sus coautores obtuvieron una llamada solución analítica exacta (que involucra una integral que solo se puede calcular numéricamente) para el modelo SIR. [6] En el caso sin configuración de dinámica vital, para, etc., corresponde a la siguiente parametrización de tiempo
por
con condiciones iniciales
where satisfies . By the transcendental equation for above, it follows that , if and .
An equivalent so-called analytical solution (involving an integral that can only be calculated numerically) found by Miller[17][18] yields
Here can be interpreted as the expected number of transmissions an individual has received by time . The two solutions are related by .
Effectively the same result can be found in the original work by Kermack and McKendrick.[4]
These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to . The equations may thus be divided through by , and the time rescaled so that the differential operator on the left-hand side becomes simply , where , i.e. . The differential equations are now all linear, and the third equation, of the form const., shows that and (and above) are simply linearly related.
A highly accurate analytic approximant of the SIR model as well as exact analytic expressions for the final values , , and were provided by Kröger and Schlickeiser,[8] so that there is no need to perform a numerical integration to solve the SIR model, to obtain its parameters from existing data, or to predict the future dynamics of an epidemics modeled by the SIR model. The approximant involves the Lambert W function which is part of all basic data visualization software such as Microsoft Excel, MATLAB, and Mathematica.
While Kendall[5] considered the so-called all-time SIR model where the initial conditions , , and are coupled through the above relations, Kermack and McKendrick[4] proposed to study the more general semi-time case, for which and are both arbitrary. This latter version, denoted as semi-time SIR model,[8] makes predictions only for future times . An analytic approximant and exact expressions for the final values are available for the semi-time SIR model as well.[9]
The SIR model with vital dynamics and constant population
Consider a population characterized by a death rate and birth rate , and where a communicable disease is spreading.[7] The model with mass-action transmission is:
for which the disease-free equilibrium (DFE) is:
In this case, we can derive a basic reproduction number:
which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that:
The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that 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 than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.
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, ; infected, ; and recovered, . The compartments used for this model consist of three classes:[4]
- is used to represent the individuals not yet infected with the disease at time t, or those susceptible to the disease of the population.
- 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.
- 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.
The flow of this model may be considered as follows:
Using a fixed population, in the three functions resolves that the value should remain constant within the simulation, if a simulation is used to solve the SIR model. Alternatively, the analytic approximant[8] can be used without performing a simulation. The model is started with values of , and . These are the number of people in the susceptible, infected and removed categories at time equals zero. If the SIR model is assumed to hold at all times, these initial conditions are not independent.[8] Subsequently, the flow model updates the three variables for every time point with set values for and . The simulation first updates the infected from the susceptible and then the removed category is updated from the infected category for the next time point (t=1). This describes the flow persons between the three categories. During an epidemic the susceptible category is not shifted with this model, changes over the course of the epidemic and so does . These variables determine the length of the epidemic and would have to be updated with each cycle.
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 and an equal fraction of people that an individual makes contact with per unit time. Then, let be the multiplication of and . This is the transmission probability times the contact rate. Besides, an infected individual makes contact with persons per unit time whereas only a fraction, of them are susceptible.Thus, we have every infective can infect susceptible persons, and therefore, the whole number of susceptibles infected by infectives per unit time is . 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 (which represents the mean recovery/death rate, or 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.[19]
Steady-state solutions
The expected duration of susceptibility will be where reflects the time alive (life expectancy) and reflects the time in the susceptible state before becoming infected, which can be simplified[20] to:
such that the number of susceptible persons is the number entering the susceptible compartment times the duration of susceptibility:
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 times the duration of infectiousness :
Other compartmental models
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).
Variaciones sobre el modelo SIR básico
The SIS model
Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections do not give immunity upon recovery from infection, and individuals become susceptible again.
We have the model:
Note that denoting with N the total population it holds that:
- .
It follows that:
- ,
i.e. the dynamics of infectious is ruled by a logistic function, so that :
It is possible to find an analytical solution to this model (by making a transformation of variables: and substituting this into the mean-field equations),[21] such that the basic reproduction rate is greater than unity. The solution is given as
- .
where is the endemic infectious population, , and . As the system is assumed to be closed, the susceptible population is then .
As a special case, one obtains the usual logistic function by assuming . This can be also considered in the SIR model with , i.e. no removal will take place. That is the SI model.[22] The differential equation system using thus reduces to:
In the long run, in the SI model, all individuals will become infected. For evaluating the epidemic threshold in the SIS model on networks see Parshani et al.[23]
The SIRD model
The Susceptible-Infectious-Recovered-Deceased model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased.[13] This model uses the following system of differential equations:
where are the rates of infection, recovery, and mortality, respectively.[24]
The MSIR model
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 is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.
To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations:
Carrier state
Some people who have had an infectious disease such as tuberculosis never completely recover and continue to 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.
The SEIR model
For many important infections, there is a significant latency period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).
Assuming that the latency period is a random variable with exponential distribution with parameter (i.e. the average latency period is ), and also assuming the presence of vital dynamics with birth rate equal to death rate (so that the total number is constant), we have the model:
We have but this is only constant because of the simplifying assumption that birth and death rates are equal; in general is a variable.
For this model, the basic reproduction number is:
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 from biologically meaningful initial conditions
it holds that:
In case of periodically varying contact rate the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:
is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).
The SEIS model
The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end.
In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model:
The MSEIR model
For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model.
The MSEIRS model
An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.
Variable contact rates
It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.
In addition, Some diseases are seasonal, such as the 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. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate
with period T equal to one year.
Thus, our model becomes
(the dynamics of recovered easily follows from ), 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:
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. 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. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.
SIR model with diffusion
Spatiotemporal compartmental models describe not the total number, but the density of susceptible/infective/recovered persons. Consequently, they also allow to model the distribution of infected persons in space. In most cases, this is done by combining the SIR model with a diffusion equation
where , and are diffusion constants. Thereby, one obtains a reaction-diffusion equation. (Note that, for dimensional reasons, the parameter has to be changed compared to the simple SIR model.) Early models of this type have been used to model the spread of the black death in Europe.[25] Extensions of this model have been used to incorporate, e.g., effects of nonpharmaceutical interventions such as social distancing.[26]
Pandemic
An SIR community based model to assess the probability for a worldwide spreading and declare pandemic has been recently developed by Valdez et al.[27]
Modelado de vacunación
The SIR model can be modified to model vaccination.[28] Typically these introduce an additional compartment to the SIR model, , for vaccinated individuals. Below are some examples.
Vaccinating newborns
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. Consider a disease for which the newborn are vaccinated (with a vaccine giving lifelong immunity) at a rate :
where is the class of vaccinated subjects. It is immediate to show that:
thus we shall deal with the long term behavior of and , for which it holds that:
In other words, if
the vaccination program is not 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 vaccinate at least 94.4% of newborns in order to eradicate the disease.
Vaccination and information
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:
In such a case the eradication condition becomes:
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.
Vaccination of non-newborns
In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows:
leading to the following eradication condition:
Pulse vaccination strategy
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:
It is easy to see that by setting I = 0 one obtains that the dynamics of the susceptible subjects is given by:
and that the eradication condition is:
La influencia de la edad: modelos estructurados por edad
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 (to limit ourselves to the susceptible-infectious-removed scheme) such that:
(where is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations:
where:
is the force of infection, which, of course, will depend, though the contact kernel on the interactions between the ages.
Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed:
but that are nonlocal for the density of susceptible newborns:
where are the fertilities of the adults.
Moreover, defining now the density of the total population one obtains:
In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demographic equilibrium the following necessary and sufficient condition linking the fertility with the mortality must hold:
and the demographic equilibrium is
automatically ensuring the existence of the disease-free solution:
A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.
Otras consideraciones dentro de los modelos epidémicos compartimentados
Vertical transmission
In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is called Vertical Transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.[29]
Vector transmission
Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic model must include both species, generally requiring many more compartments than a model for direct transmission.[29][30]
Others
Other occurrences which may need to be considered when modeling an epidemic include things such as the following:[29]
- Non-homogeneous mixing
- Variable infectivity
- Distributions that are spatially non-uniform
- Diseases caused by macroparasites
Modelos de epidemia deterministas versus estocásticos
It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.[31]
To be more precise, these models are only valid in the thermodynamic limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.
One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts.[32] Stochastic epidemic models have been studied on different networks[33][34][35] and more recently applied to the COVID-19 pandemic.[36]
Distribución eficiente de vacunas
A method for efficient vaccination approach, via vaccinating a small fraction population called acquaintance immunization has been developed by Cohen et al.[37] An alternative method based on identifying and vaccination mainly spreaders has been developed by Liu et al[38]
Ver también
- Mathematical modelling in epidemiology
- Modifiable areal unit problem
- Next-generation matrix
- Risk assessment
- Attack rate
- List of COVID-19 simulation models
Referencias
- ^ Ross, Ronald (1 February 1916). "An application of the theory of probabilities to the study of a priori pathometry.—Part I". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 92 (638): 204–230. Bibcode:1916RSPSA..92..204R. doi:10.1098/rspa.1916.0007.
- ^ Ross, Ronald; Hudson, Hilda (3 May 1917). "An application of the theory of probabilities to the study of a priori pathometry.—Part II". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 93 (650): 212–225. Bibcode:1917RSPSA..93..212R. doi:10.1098/rspa.1917.0014.
- ^ Ross, Ronald; Hudson, Hilda (1917). "An application of the theory of probabilities to the study of a priori pathometry.—Part III". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 89 (621): 225–240. Bibcode:1917RSPSA..93..225R. doi:10.1098/rspa.1917.0015.
- ^ a b c d Kermack, W. O.; McKendrick, A. G. (1927). "A Contribution to the Mathematical Theory of Epidemics". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 115 (772): 700–721. Bibcode:1927RSPSA.115..700K. doi:10.1098/rspa.1927.0118.
- ^ a b Kendall, D. G. (1956). "Deterministic and stochastic epidemics in closed populations". Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability: Contributions to Biology and Problems of Health. 4: 149–165. doi:10.1525/9780520350717-011. ISBN 9780520350717. MR 0084936. Zbl 0070.15101.
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- ^ (p. 19) The SI Model
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- ^ The first and second differential equations are transformed and brought to the same form as for the SIR model above.
- ^ Noble, J.V. (1974). "Geographic and temporal development of plagues". Nature. 250 (5469): 726–729. Bibcode:1974Natur.250..726N. doi:10.1038/250726a0. PMID 4606583. S2CID 4210869.
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- ^ a b c Brauer, F.; Castillo-Chávez, C. (2001). Mathematical Models in Population Biology and Epidemiology. NY: Springer. ISBN 0-387-98902-1.
- ^ For more information on this type of model see Anderson, R. M., ed. (1982). Population Dynamics of Infectious Diseases: Theory and Applications. London-New York: Chapman and Hall. ISBN 0-412-21610-8.
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- ^ May, Robert M.; Lloyd, Alun L. (2001-11-19). "Infection dynamics on scale-free networks". Physical Review E. 64 (6): 066112. Bibcode:2001PhRvE..64f6112M. doi:10.1103/PhysRevE.64.066112. PMID 11736241.
- ^ Pastor-Satorras, Romualdo; Vespignani, Alessandro (2001-04-02). "Epidemic Spreading in Scale-Free Networks". Physical Review Letters. 86 (14): 3200–3203. arXiv:cond-mat/0010317. Bibcode:2001PhRvL..86.3200P. doi:10.1103/PhysRevLett.86.3200. hdl:2117/126209. PMID 11290142. S2CID 16298768.
- ^ Newman, M. E. J. (2002-07-26). "Spread of epidemic disease on networks". Physical Review E. 66 (1): 016128. arXiv:cond-mat/0205009. Bibcode:2002PhRvE..66a6128N. doi:10.1103/PhysRevE.66.016128. PMID 12241447. S2CID 15291065.
- ^ Wong, Felix; Collins, James J. (2020-11-02). "Evidence that coronavirus superspreading is fat-tailed". Proceedings of the National Academy of Sciences. 117 (47): 29416–29418. Bibcode:2020PNAS..11729416W. doi:10.1073/pnas.2018490117. ISSN 0027-8424. PMC 7703634. PMID 33139561. S2CID 226242440.
- ^ R Cohen, S Havlin, D Ben-Avraham (2003). "Efficient immunization strategies for computer networks and populations". Physical Review Letters. 91 (24): 247901. arXiv:cond-mat/0207387. Bibcode:2003PhRvL..91x7901C. doi:10.1103/PhysRevLett.91.247901. PMID 14683159. S2CID 919625.CS1 maint: multiple names: authors list (link)
- ^ Y Liu, H Sanhedrai, GG Dong, LM Shekhtman, F Wang, SV Buldyrev, S. Havlin (2021). "Efficient network immunization under limited knowledge". National Science Review. 8 (1). arXiv:2004.00825. doi:10.1093/nsr/nwaa229.CS1 maint: multiple names: authors list (link)
Otras lecturas
- May, Robert M.; Anderson, Roy M. (1991). Infectious diseases of humans: dynamics and control. Oxford: Oxford University Press. ISBN 0-19-854040-X.
- Vynnycky, E.; White, R. G., eds. (2010). An Introduction to Infectious Disease Modelling. Oxford: Oxford University Press. ISBN 978-0-19-856576-5.
- Capasso, Vincenzo (2008). Mathematical Structures of Epidemic Systems. 2nd Printing. Heidelberg: Springer. ISBN 978-3-540-56526-0.
enlaces externos
- SIR model: Online experiments with JSXGraph
- "Simulating an epidemic". 3Blue1Brown. March 27, 2020 – via YouTube.