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The influenza pandemic preparedness planning tool InfluSim

Background

Preparedness against pandemic influenza has become a high priority public health issue and many countries that have pandemic preparedness plans. For the design of such plans, mathematical models and computer simulations play an essential role because they allow to predict and compare the effects of different intervention strategies. The outstanding significance of the tools for purposes of intervention optimization is limited by the fact that they cannot maximize realism, generality and precision at the same time. Public health planners, on the other hand, wish to have an optimal combination of these properties, because they need to formulate intervention strategies which can be generalized into recommendations, but are sufficiently realistic and precise to satisfy public health requirements.

Published influenza models which came into application, are represented by two extremes: generalized but over-simplified models without dynamic structure which are publicly available (e.g.), and complex computer simulations which are specifically adjusted to real conditions and/or are not publicly available (e.g.). The complexity of the latter simulations, however, is not necessary for a reliable description of infection dynamics in large populations. A minimum requirement for a pandemic influenza planning tool is a dynamic modelling structure which allows investigation of time-dependent variables like incidence, height of the epidemic peak, antiviral availability etc. The tool should, on the other hand, be adjustable to local conditions to adequately support the pandemic preparedness plans of different countries which involve considerably different assumptions (Table 1).

Here we describe a publicly available influenza pandemic preparedness planning tool which is designed to meet the requirements in preparedness planning. It is based on an explicitly formulated dynamic system which allows addressing time-dependent factors. It is sufficiently flexible to evaluate the impact of most candidate interventions and to consider local conditions like demographic and economic factors, contact patterns or constraints within the public health system. In subsequent papers we will also provide examples and applications of this model for various interventions, like antiviral treatment and social distancing measures.

Implementation

The model is based on a system of 1,081 differential equations which extend the classic SEIR model. Demographic parameters reflect the situation in Germany in 2005, but can be adjusted to other countries. Epidemiologic and clinic values were taken from the literature (see Tables 1, 2, 3, 4, 5, 6 and the sources quoted there). Pre-set values can be varied by sliders and input fields to make different assumptions on the transmissibility and clinical severity of a new pandemic strain, to change the costs connected to medical treatment or work loss, or to simply apply the simulation to different demographic settings. Model properties can be summarized as follows. The mathematical formulation of this model is presented in detail in the online supporting material. The corresponding source code, programmed in Java, and further information can be downloaded from.

According to the German National Pandemic Preparedness Plan, the total population is divided in age classes, each of which is subdivided into individuals of low and high risk (Table 2). Transmission between these age classes is based on a contact matrix (Table 3) which is scaled such that the model with standard parameter values yields a given basic reproduction number R0. Values for the R0 associated with an influenza strain with pandemic potential are suggested to lie between 2 and 3. This value is higher than the effective reproduction number which has been estimated to be slightly lower than 2. As a standard parameter, we use R0 = 2.5 which means that cases infect on average 2.5 individuals if everybody is susceptible and if no interventions are performed.

Susceptible individuals who become infected, incubate the infection, then become fully contagious and finally develop protective immunity (Table 4). A fraction of cases remains asymptomatic; others become moderately sick or clinically ill (i.e. they need medical help). Depending on the combination of age and risk group, a fraction of the clinically ill cases needs to be hospitalized, and an age-dependent fraction of hospitalized cases may die from the disease (Table 5). This partitioning of the cases into four categories allows combining the realistic description of the transmission dynamics with an easy calculation of the resources consumed during an outbreak. The degree and duration of contagiousness of a patient depend on the course of the disease; the latter furthermore depends on the age of the patient (Table 5). Passing through the incubation and contagious period is modelled in several stages which allows for realistic distributions of the sojourn times (Table 4). The last two stages of the incubation period are used as early infectious period during which the patient can already spread the disease. Infectiousness is highest after onset of symptoms and thereafter declines geometrically (Table 6). Clinically ill patients seek medical help on average one day after onset of symptoms. Very sick patients are advised to withdraw to their home until their disease is over, whereas extremely sick patients need to be hospitalized and may die from the disease (Table 4). After the end of their contagious period, clinically ill patients go through a convalescent period before they can resume their ordinary life and go back to work (Table 4).

Results

We provide some examples of model output of InfluSim, version 2.0, by means of four sensitivity analyses; further investigations will be presented elsewhere. Figure 1 shows the graphical user interface of the software which is divided into input and output windows. The user may set new values in the input fields or move sliders to almost simultaneously obtain new results for the course of an epidemic in a given population. Figures 2A and 2B show pandemic waves which result from varying the basic reproduction number from 1.5 to 4.0. Using the standard parameter values as given in Tables 2, 3, 4, 5, 6 and omitting all interventions in a town of 100,000 inhabitants results in a pandemic wave which lasts for about ten weeks (Figure 2A, with R0 = 2.5). The peak of the pandemic wave is reached after six to seven weeks, with a daily incidence of up to 2,340 influenza patients seeking medical help, with up to 280 hospital beds occupied by influenza cases and with up to 14,000 out of 60,000 working adults unable to go to work because of illness or convalescence. These results depend on the assumptions concerning the yet unknown contagiousness and pathogenicity of the virus. Figures 2C and 2D show how the shape of the curves depends on the course of contagiousness: the pandemic wave proceeds relative slowly if the contagiousness does not change during the infectious period (x50 = 50%), but proceeds quickly if the contagiousness is highest after onset of symptoms and decreases thereafter (x50 > 50%).

Discussion and Conclusion

The influenza pandemic preparedness planning tool InfluSim stands between simple spreadsheet models and sophisticated stochastic computer simulations. It describes a pandemic wave within a homogeneously mixing population like a town or city, but surprisingly produces the same dynamics as individual-based simulations which explicitly consider geographic spread through the US (cf. and with Figure 2 using R0 = 2). Similar observations were made with a simple deterministic compartmental model. Stochastic models are known to behave quasi-deterministically when the simulated population becomes very large.

A further reason for the congruence of complex stochastic and simple deterministic models must lie in the incredibly quick way in which pandemic influenza spreads geographically. Unless being controlled at the place of origin, a pandemic starting in a far-off country will lead to multiple introductions into the large industrialized nations where it can be expected to quickly spread to neighbouring towns and to rural areas. The large populations which have to be considered susceptible to a pandemic virus and the quick geographic spread tend to diminish the differences between the results of sophisticated individual-based and simple deterministic models.

However, a deterministic model like InfluSim cannot reliably represent effects originating from stochasticity, from effects in small populations, or from heterogeneities. Examples are: (i) a geographically limited spread and fairly effective control measures can imply that the epidemic affects only a small population and thus, may be strongly influenced by stochastic events; (ii) transmission which predominantly occurs in households or hospitals, or which is driven by other substantial features of the contact network is not in agreement with the assumption of homogeneous mixing in the deterministic model cannot reliably predict the spread of infection. In particular, (iii) super-spreading events can substantially change the course of an epidemic compared to the deterministic prediction. Apart from such factors, the predictability of intervention success is generally subject to uncertainties in the choice of parameter values, demanding additional efforts like Bayesian approaches to evaluate the reliability of predictions.

Pandemic preparedness plans must consider constraints and capacities of locally operating public health systems. The time-dependent solutions of InfluSim allow assessing peak values of the relevant variables, such as outpatients, hospitalizations and deaths. Various interventions may be combined to find optimal ways to reduce the total number of cases, to lower the peak values or to delay the peak, hoping that at least part of the population may benefit from a newly developed vaccine.

Special care was taken when implementing a variety of pharmaceutical and non-pharmaceutical interventions which will be discussed in subsequent papers. Despite its comprehensible structure, the model does not suffer from over-simplifications common to usual compartment models. Instead of implicitly using exponentially distributed sojourn times, we have implemented realistically distributed delays. For example, the model considers that individuals may transmit infection before onset of symptoms, and that some cases may remain asymptomatic, but still infecting others. Such features have serious implications for the success of targeted control measures.

InfluSim is freely accessible, runs on a regular desktop computer and produces results within a second after changing parameter values. The user-friendly interface and the ease at which results can be generated make this program a useful public health planning tool. Although we have taken care of providing a bug-free program, including the source code, the user is encouraged to treat results with due caution, to test it, and to participate in bug-reports and discussions on the open-source platform which also provides regular updates of InfluSim.

Availability and requirements

Project name: InfluSim version 2.0

Project home page:

Sourceforge:

Operating systems: Platform independent

Programming language: Java

Other requirements: e.g. Java 1.5 or higher

License: CPL

Any restrictions to use by non-academics: none

Competing interests

The author(s) declare that they have no competing interests.

Authors' contributions

ME developed the model, MS designed the software, HPD wrote the manuscript and SOB formulated the public health requirements of the software. All authors read and approved the final manuscript.

Differential equation model describing the transmission dynamics

Susceptible individuals

S˙a,r=−λa(t)Sa,r

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Infected individuals who incubate the infection

E˙1,a,r=λa(t)Sa,r−δE1,a,rE˙k,a,r=δ(Ek−1,a,r−Ek,a,r)for k=2,...,n

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Asymptomatic infectious individuals

A˙1,a=δca,r(A)En,a,r−γA,aA1,aA˙k,a=γA,a(Ak−1,a−Ak,a)for k=2,...,m

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Moderately sick individuals

M˙1,a=δca,r(M)En,a,r−γM,aM1,aM˙k,a=γM,a(Mk−1,a−Mk,a)for k=2,...,m

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Very sick individuals who have not yet visited a doctor

V˙1,a=δca,r(V)En,a,r−(γV,a,U+α)V1,aV˙k,a=γV,a,U(Vk−1,a−Vk,a)−αVk,afor k=2,...,m

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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f7aHjabdAfawnaaBaaaleaacqWGRbWAcqGGSaalcqWGHbqyaeqaaaaakeaacqqGMbGzcqqGVbWBcqqGYbGCcqqGGaaicqWGRbWAcqGH9aqpcqaIYaGmcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGTbqBaaaaaa@8503@

Treated very sick individuals

W˙1,a,T=α(fV(t)V1,a+fX(t)fHX1,a)−γV,a,TW1,a,TW˙k,a,T=α(fV(t)Vk,a+fX(t)fHXk,a)+γV,a,T(Wk−1,a,T−Wk,a,T)for k=2,...,ma,TW˙k,a,T=γV,a,T(Wk−1,a,T−Wk,a,T)for k=ma,T+1,...,m

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeWacaaabaGafm4vaCLbaiaadaWgaaWcbaGaeGymaeJaeiilaWIaemyyaeMaeiilaWIaemivaqfabeaakiabg2da9GGaciab=f7aHnaabmaabaGaemOzay2aaSbaaSqaaiabdAfawbqabaGccqGGOaakcqWG0baDcqGGPaqkcqWGwbGvdaWgaaWcbaGaeGymaeJaeiilaWIaemyyaegabeaakiabgUcaRiabdAgaMnaaBaaaleaacqWGybawaeqaaOGaeiikaGIaemiDaqNaeiykaKIaemOzay2aaSbaaSqaaiabdIeaibqabaGccqWGybawdaWgaaWcbaGaeGymaeJaeiilaWIaemyyaegabeaaaOGaayjkaiaawMcaaiabgkHiTiab=n7aNnaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqycqGGSaalcqWGubavaeqaaOGaem4vaC1aaSbaaSqaaiabigdaXiabcYcaSiabdggaHjabcYcaSiabdsfaubqabaaakeaaaeaacuWGxbWvgaGaamaaBaaaleaacqWGRbWAcqGGSaalcqWGHbqycqGGSaalcqWGubavaeqaaOGaeyypa0Jae8xSde2aaeWaaeaacqWGMbGzdaWgaaWcbaGaemOvayfabeaakiabcIcaOiabdsha0jabcMcaPiabdAfawnaaBaaaleaacqWGRbWAcqGGSaalcqWGHbqyaeqaaOGaey4kaSIaemOzay2aaSbaaSqaaiabdIfaybqabaGccqGGOaakcqWG0baDcqGGPaqkcqWGMbGzdaWgaaWcbaGaemisaGeabeaakiabdIfaynaaBaaaleaacqWGRbWAcqGGSaalcqWGHbqyaeqaaaGccaGLOaGaayzkaaGaey4kaSIae83SdC2aaSbaaSqaaiabdAfawjabcYcaSiabdggaHjabcYcaSiabdsfaubqabaGcdaqadaqaaiabdEfaxnaaBaaaleaacqWGRbWAcqGHsislcqaIXaqmcqGGSaalcqWGHbqycqGGSaalcqWGubavaeqaaOGaeyOeI0Iaem4vaC1aaSbaaSqaaiabdUgaRjabcYcaSiabdggaHjabcYcaSiabdsfaubqabaaakiaawIcacaGLPaaaaeaacqqGMbGzcqqGVbWBcqqGYbGCcqqGGaaicqWGRbWAcqGH9aqpcqaIYaGmcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGTbqBdaWgaaWcbaGaemyyaeMaeiilaWIaemivaqfabeaaaOqaaiqbdEfaxzaacaWaaSbaaSqaaiabdUgaRjabcYcaSiabdggaHjabcYcaSiabdsfaubqabaGccqGH9aqpcqWFZoWzdaWgaaWcbaGaemOvayLaeiilaWIaemyyaeMaeiilaWIaemivaqfabeaakmaabmaabaGaem4vaC1aaSbaaSqaaiabdUgaRjabgkHiTiabigdaXiabcYcaSiabdggaHjabcYcaSiabdsfaubqabaGccqGHsislcqWGxbWvdaWgaaWcbaGaem4AaSMaeiilaWIaemyyaeMaeiilaWIaemivaqfabeaaaOGaayjkaiaawMcaaaqaaiabbAgaMjabb+gaVjabbkhaYjabbccaGiabdUgaRjabg2da9iabd2gaTnaaBaaaleaacqWGHbqycqGGSaalcqWGubavaeqaaOGaey4kaSIaeGymaeJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemyBa0gaaaaa@E514@

Untreated very sick individuals

W˙1,a,U=α(1−fV(t))V1,a−γV,a,UW1,a,UW˙k,a,U=α(1−fV(t))Vk,a+γV,a,U(Wk−1,a,U−Wk,a,U)for k=2,...,ma,TW˙k,a,U=αVk,a+γV,a,U(Wk−1,a,U−Wk,a,U)for k=ma,T+1,...,m

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeWacaaabaGafm4vaCLbaiaadaWgaaWcbaGaeGymaeJaeiilaWIaemyyaeMaeiilaWIaemyvaufabeaakiabg2da9GGaciab=f7aHnaabmaabaGaeGymaeJaeyOeI0IaemOzay2aaSbaaSqaaiabdAfawbqabaGccqGGOaakcqWG0baDcqGGPaqkaiaawIcacaGLPaaacqWGwbGvdaWgaaWcbaGaeGymaeJaeiilaWIaemyyaegabeaakiabgkHiTiab=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n7aNnaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqycqGGSaalcqWGvbqvaeqaaOWaaeWaaeaacqWGxbWvdaWgaaWcbaGaem4AaSMaeyOeI0IaeGymaeJaeiilaWIaemyyaeMaeiilaWIaemyvaufabeaakiabgkHiTiabdEfaxnaaBaaaleaacqWGRbWAcqGGSaalcqWGHbqycqGGSaalcqWGvbqvaeqaaaGccaGLOaGaayzkaaaabaGaeeOzayMaee4Ba8MaeeOCaiNaeeiiaaIaem4AaSMaeyypa0JaemyBa02aaSbaaSqaaiabdggaHjabcYcaSiabdsfaubqabaGccqGHRaWkcqaIXaqmcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGTbqBaaaaaa@D3FE@

Extremely sick individuals who have not yet visited a doctor

X˙1,a=δca,r(X)En,a,r−(γV,a+α+τa)X1,aX˙k,a=γV,a(Xk−1,a−Xk,a)−(α+τa)Xk,afor k=2,...,m

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaaeaqabeaacuWGybawgaGaamaaBaaaleaacqaIXaqmcqGGSaalcqWGHbqyaeqaaOGaeyypa0dcciGae8hTdqMaem4yam2aaSbaaSqaaiabdggaHjabcYcaSiabdkhaYbqabaGcdaqadaqaaiabdIfaybGaayjkaiaawMcaaiabdweafnaaBaaaleaacqWGUbGBcqGGSaalcqWGHbqycqGGSaalcqWGYbGCaeqaaOGaeyOeI0YaaeWaaeaacqWFZoWzdaWgaaWcbaGaemOvayLaeiilaWIaemyyaegabeaakiabgUcaRiab=f7aHjabgUcaRiab=r8a0naaBaaaleaacqWGHbqyaeqaaaGccaGLOaGaayzkaaGaemiwaG1aaSbaaSqaaiabigdaXiabcYcaSiabdggaHbqabaaakeaacuWGybawgaGaamaaBaaaleaacqWGRbWAcqGGSaalcqWGHbqyaeqaaOGaeyypa0Jae83SdC2aaSbaaSqaaiabdAfawjabcYcaSiabdggaHbqabaGcdaqadaqaaiabdIfaynaaBaaaleaacqWGRbWAcqGHsislcqaIXaqmcqGGSaalcqWGHbqyaeqaaOGaeyOeI0IaemiwaG1aaSbaaSqaaiabdUgaRjabcYcaSiabdggaHbqabaaakiaawIcacaGLPaaacqGHsisldaqadaqaaiab=f7aHjabgUcaRiab=r8a0naaBaaaleaacqWGHbqyaeqaaaGccaGLOaGaayzkaaGaemiwaG1aaSbaaSqaaiabdUgaRjabcYcaSiabdggaHbqabaaaaOqaaiabbAgaMjabb+gaVjabbkhaYjabbccaGiabdUgaRjabg2da9iabikdaYiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabd2gaTbaaaaa@8AA2@

Hospitalized and treated cases

H˙1,a,T=αfX(t)(1−fH)X1,a−(γV,a,T+τa)H1,a,TH˙k,a,T=αfX(t)(1−fH)Xk,a+γV,a,T(Hk−1,a,T−Hk,a,T)−τaHk,a,Tfor k=2,...ma,TH˙k,a,T=γV,a,T(Hk−1,a,T−Hk,a,T)−τaHk,a,Tfor k=ma,T+1,...,m

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeWacaaabaGafmisaGKbaiaadaWgaaWcbaGaeGymaeJaeiilaWIaemyyaeMaeiilaWIaemivaqfabeaakiabg2da9GGaciab=f7aHjabdAgaMnaaBaaaleaacqWGybawaeqaaOGaeiikaGIaemiDaqNaeiykaKYaaeWaaeaacqaIXaqmcqGHsislcqWGMbGzdaWgaaWcbaGaemisaGeabeaaaOGaayjkaiaawMcaaiabdIfaynaaBaaaleaacqaIXaqmcqGGSaalcqWGHbqyaeqaaOGaeyOeI0YaaeWaaeaacqWFZoWzdaWgaaWcbaGaemOvayLaeiilaWIaemyyaeMaeiilaWIaemivaqfabeaakiabgUcaRiab=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r8a0naaBaaaleaacqWGHbqyaeqaaOGaemisaG0aaSbaaSqaaiabdUgaRjabcYcaSiabdggaHjabcYcaSiabdsfaubqabaaakeaacqqGMbGzcqqGVbWBcqqGYbGCcqqGGaaicqWGRbWAcqGH9aqpcqWGTbqBdaWgaaWcbaGaemyyaeMaeiilaWIaemivaqfabeaakiabgUcaRiabigdaXiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabd2gaTbaaaaa@EBD5@

Hospitalized, but untreated cases

H˙1,a,U=α(1−fX(t))X1,a−(γV,a,U+τa)H1,a,UH˙k,a,U=α(1−fX(t))Xk,a+γV,a,U(Hk−1,a,U−Hk,a,U)−τaHk,a,Ufor k=2,...,ma,TH˙k,a,U=αXk,a+γV,a,U(Hk−1,a,U−Hk,a,U)−τaHk,a,Ufor k=ma,T+1,...,m

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeWacaaabaGafmisaGKbaiaadaWgaaWcbaGaeGymaeJaeiilaWIaemyyaeMaeiilaWIaemyvaufabeaakiabg2da9GGaciab=f7aHnaabmaabaGaeGymaeJaeyOeI0IaemOzay2aaSbaaSqaaiabdIfaybqabaGccqGGOaakcqWG0baDcqGGPaqkaiaawIcacaGLPaaacqWGybawdaWgaaWcbaGaeGymaeJaeiilaWIaemyyaegabeaakiabgkHiTmaabmaabaGae83SdC2aaSbaaSqaaiabdAfawjabcYcaSiabdggaHjabcYcaSiabdwfavbqabaGccqGHRaWkcqWFepaDdaWgaaWcbaGaemyyaegabeaaaOGaayjkaiaawMcaaiabdIeainaaBaaaleaacqaIXaqmcqGGSaalcqWGHbqycqGGSaalcqWGvbqvaeqaaaGcbaaabaGafmisaGKbaiaadaWgaaWcbaGaem4AaSMaeiilaWIaemyyaeMaeiilaWIaemyvaufabeaakiabg2da9iab=f7aHnaabmaabaGaeGymaeJaeyOeI0IaemOzay2aaSbaaSqaaiabdIfaybqabaGccqGGOaakcqWG0baDcqGGPaqkaiaawIcacaGLPaaacqWGybawdaWgaaWcbaGaem4AaSMaeiilaWIaemyyaegabeaakiabgUcaRiab=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@EEFC@

Contact matrix

For the mixing of the age classes, we employ a who-acquires-infection-from whom matrix M=(mas,ai)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHnbqtcqGH9aqpcqGGOaakcqWHTbqBdaWgaaWcbaGaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaGccqGGPaqkaaa@38D8@ which gives the relative frequency of contacts of infective individuals of age ai with other people of age as. In this paper, we assume bi-directional contacts (e.g. children have the same total number of contacts with adults as adults with children). Multiplication of this matrix with an appropriate constant scaling factor κ (see below) results in the matrix of crude contact rates βas,ai=κ mas,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaGccqGH9aqpcqWF6oWAcqqGGaaicqWHTbqBdaWgaaWcbaGaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaaaaa@40F7@.

Contagiousness of the different types of disease

In the absence of interventions, we have to multiply these contact rates with the contagiousness factors bL, bA, bM and bV to obtain the effective contact rates:

βL,as,ai=bLβas,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemitaWKaeiilaWIaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaGccqGH9aqpcqWGIbGydaWgaaWcbaGaemitaWeabeaakiab=j7aInaaBaaaleaacqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@435D@ during the early infectious period,

βA,as,ai=bAβas,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyqaeKaeiilaWIaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaGccqGH9aqpcqWGIbGydaWgaaWcbaGaemyqaeeabeaakiab=j7aInaaBaaaleaacqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@4331@ of asymptomatic cases,

βM,as,ai=bMβas,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyta0KaeiilaWIaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaGccqGH9aqpcqWGIbGydaWgaaWcbaGaemyta0eabeaakiab=j7aInaaBaaaleaacqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@4361@ of moderately sick cases,

βV,as,ai=bVβas,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemOvayLaeiilaWIaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaGccqGH9aqpcqWGIbGydaWgaaWcbaGaemOvayfabeaakiab=j7aInaaBaaaleaacqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@4385@ of (untreated) very sick cases.

Day care centres and schools

To assess the effect of day care centre and school closing on the transmission of an infectious disease, we have to first make an assumption on what fraction rsch of the contacts among healthy children who are in the same age class occurs in day care centres and schools. The contact rates between very sick or hospitalized children (who do not attend day care centre or school) and other children need, therefore, be reduced to β′V,as,ai(t)=(1−rsch)βV,as,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqbamaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iabcIcaOiabigdaXiabgkHiTiabdkhaYnaaBaaaleaacqWGZbWCcqWGJbWycqWGObaAaeqaaOGaeiykaKIae8NSdi2aaSbaaSqaaiabdAfawjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaaaa@4F5A@ (contact rate between healthy and very sick children in the same age class, i.e. ai = as).

As very sick children have to be taken care of by adults at home or in hospital, their contact rate to adults increases by a factor β′V,as,ai(t)=βV,as,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqbamaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iab=j7aInaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@4611@FHC (contact rate between very sick children of age ai and adults of age as).

Contacts between very sick children and other children in a higher or lower age class remain unchanged: β′V,as,ai(t)=βV,as,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqbamaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iab=j7aInaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@4611@ (contact rate between healthy children of age as and very sick children of a different age ai).

Closing of day care centres and schools

Closing day care centres and schools at time t will not necessarily prevent all the contacts that would have happened with other children. During the closing of schools and day care centres, the contact rates between susceptible children of age as and infected children of age ai who are in their late incubation period (βL,as,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemitaWKaeiilaWIaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaaaaa@3731@), who are asymptomatic (βA,as,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyqaeKaeiilaWIaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaaaaa@371B@), or who are moderately sick (βM,as,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyta0KaeiilaWIaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaaaaa@3733@) are reduced by the factor rsch if the children are in the same age class:

β′L,as,ai(t)={βL,as,ai(1−rsch)1sch(t)if as=aiβL,as,aiif as≠ai,β′A,as,ai(t)={βA,as,ai(1−rsch)1sch(t)if as=aiβA,as,aiif as≠ai,β′M,as,ai(t)={βM,as,ai(1−rsch)1sch(t)if as=aiβM,as,aiif as≠ai.

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaGGaciqb=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j7aInaaBaaaleaacqWGmbatcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaaaOqaaiabdMgaPjabdAgaMjabbccaGiabdggaHnaaBaaaleaacqWGZbWCaeqaaOGaeyiyIKRaemyyae2aaSbaaSqaaiabdMgaPbqabaGccqGGSaalaaaacaGL7baaaeaacuWFYoGygaqbamaaBaaaleaacqWGbbqqcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9maaceqabaqbaeaabiGaaaqaaiab=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@137C@

where 1sch (t) is a function which indicates when schools and day care centres are opened or closed:

1sch(t)={1⥄⥄⥄while day care centres and schools are closed0⥄⥄⥄while day care centres and schools are opened.

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B52D@

While day care centres and schools are closed, children (age ai) need adult supervision at home. Their contact with susceptible adults (age as) increases by the "child care factor" FCC:

β′L,as,ai(t)=βL,as,ai(FCC)1sch(t),β′A,as,ai(t)=βA,as,ai(FCC)1sch(t),β′M,as,ai(t)=βM,as,ai(FCC)1sch(t),

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaGGaciqb=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@A387@

Child care at home also increases the exposure of healthy children (age as) to contagious adults (age ai):

β′L,as,ai(t)=βL,as,ai(FCC)1sch(t),β′A,as,ai(t)=βA,as,ai(FCC)1sch(t),β′M,as,ai(t)=βM,as,ai(FCC)1sch(t),β′V,as,ai(t)=βV,as,ai(FCC)1sch(t).

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaGGaciqb=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j7aIzaafaWaaSbaaSqaaiabdAfawjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKIaeyypa0Jae8NSdi2aaSbaaSqaaiabdAfawjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOWaaeWaaeaacqWGgbGrdaWgaaWcbaGaem4qamKaem4qameabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeCymaeZaaSbaaWqaaiabdohaZjabdogaJjabdIgaObqabaWccqGGOaakcqWG0baDcqGGPaqkaaGccqGGUaGlaaaa@CB59@

Cancelling of mass gathering events

Cancelling mass gathering events effects only the contacts of adults who are healthy enough to attend such events. Assuming that such an intervention at time t reduces contacts by a fraction rmass, we get for all contacts between susceptible adults of age as and infectious adults of age ai the following contact rates:

β′L,as,ai(t)=βL,as,ai(1−rmass)1mass(t),β′A,as,ai(t)=βA,as,ai(1−rmass)1mass(t),β′M,as,ai(t)=βM,as,ai(1−rmass)1mass(t).

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakqaabeqaaGGaciqb=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j7aInaaBaaaleaacqWGnbqtcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabigdaXiabgkHiTiabdkhaYnaaBaaaleaacqWGTbqBcqWGHbqycqWGZbWCcqWGZbWCaeqaaOGaeiykaKYaaWbaaSqabeaacqWHXaqmdaWgaaadbaGaemyBa0MaemyyaeMaem4CamNaem4CamhabeaaliabcIcaOiabdsha0jabcMcaPaaakiabc6caUaaaaa@B94E@

where 1mass (t) is a function which indicates when mass gathering events are possible or when they are closed:

1mass(t)={1⥄⥄⥄while mass gathering events are forbidden0⥄⥄⥄while mass gathering events are allowed.

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As contacts with adults who are too sick to attend such mass gathering events cannot be prevented by this measure it is

β′V,as,ai(t)=βV,as,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaqbamaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iab=j7aInaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@4611@.

General reduction of contacts

During some time in the epidemic, the general population may effectively reduce contacts which can be a result of wearing facial masks, increasing "social distance", adopting improved measures of "respiratory hygiene" or simply of a general change in behaviour. This will be implemented in the program by reducing the contacts of susceptible individuals at that time t by factor rgen (t). The adjusted contact rates are:

β″L,as,ai(t)=β′L,as,ai(t)(1−rgen)1gen(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGbamaaBaaaleaacqWGmbatcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaafaWaaSbaaSqaaiabdYeamjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKYaaeWaaeaacqaIXaqmcqGHsislcqWGYbGCdaWgaaWcbaGaem4zaCMaemyzauMaemOBa4gabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeCymaeZaaSbaaWqaaiabdEgaNjabdwgaLjabd6gaUbqabaWccqGGOaakcqWG0baDcqGGPaqkaaaaaa@5AC1@ for cases in the late incubation period,

β″A,as,ai(t)=β′A.as,ai(t)(1−rgen)1gen(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGbamaaBaaaleaacqWGbbqqcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaafaWaaSbaaSqaaiabdgeabjabc6caUiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKYaaeWaaeaacqaIXaqmcqGHsislcqWGYbGCdaWgaaWcbaGaem4zaCMaemyzauMaemOBa4gabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeCymaeZaaSbaaWqaaiabdEgaNjabdwgaLjabd6gaUbqabaWccqGGOaakcqWG0baDcqGGPaqkaaaaaa@5A99@ for asymptomatic cases,

β″M,as,ai(t)=β′M.as,ai(t)(1−rgen)1gen(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGbamaaBaaaleaacqWGnbqtcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaafaWaaSbaaSqaaiabd2eanjabc6caUiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKYaaeWaaeaacqaIXaqmcqGHsislcqWGYbGCdaWgaaWcbaGaem4zaCMaemyzauMaemOBa4gabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeCymaeZaaSbaaWqaaiabdEgaNjabdwgaLjabd6gaUbqabaWccqGGOaakcqWG0baDcqGGPaqkaaaaaa@5AC9@ for moderately sick cases,

β″V,as,ai(t)=β′V,as,ai(t)(1−rgen)1gen(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGbamaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaafaWaaSbaaSqaaiabdAfawjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKYaaeWaaeaacqaIXaqmcqGHsislcqWGYbGCdaWgaaWcbaGaem4zaCMaemyzauMaemOBa4gabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeCymaeZaaSbaaWqaaiabdEgaNjabdwgaLjabd6gaUbqabaWccqGGOaakcqWG0baDcqGGPaqkaaaaaa@5AE9@ for very sick cases,

where 1gen (t) is a function which indicates when the population reduces their contacts:

1gen(t)={1⥄⥄⥄while the population reduces their contacts0⥄⥄⥄while the population behaves as usual.

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@AADD@

Partial isolation of cases

If cases are (partly) isolated, their contact rates are reduced by factors (1−risoM)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaaiabigdaXiabgkHiTiabdkhaYnaaBaaaleaacqWGPbqAcqWGZbWCcqWGVbWBdaWgaaadbaGaemyta0eabeaaaSqabaaakiaawIcacaGLPaaaaaa@3741@, (1−risoV)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaaiabigdaXiabgkHiTiabdkhaYnaaBaaaleaacqWGPbqAcqWGZbWCcqWGVbWBdaWgaaadbaGaemOvayfabeaaaSqabaaakiaawIcacaGLPaaaaaa@3753@ and (1−risoH)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqadaqaaiabigdaXiabgkHiTiabdkhaYnaaBaaaleaacqWGPbqAcqWGZbWCcqWGVbWBdaWgaaadbaGaemisaGeabeaaaSqabaaakiaawIcacaGLPaaaaaa@3737@, respectively, resulting in contact rates

β‴M,as,ai(t)=β″M,as,ai(t)(1−risoM)1iso(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGeamaaBaaaleaacqWGnbqtcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaagaWaaSbaaSqaaiabd2eanjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKYaaeWaaeaacqaIXaqmcqGHsislcqWGYbGCdaWgaaWcbaGaemyAaKMaem4CamNaem4Ba82aaSbaaWqaaiabd2eanbqabaaaleqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqWHXaqmdaWgaaadbaGaemyAaKMaem4CamNaem4Ba8gabeaaliabcIcaOiabdsha0jabcMcaPaaaaaa@5C71@ for moderately sick cases,

β‴V,as,ai(t)=β″V,as,ai(t)(1−risoV)1iso(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGeamaaBaaaleaacqWGwbGvcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaagaWaaSbaaSqaaiabdAfawjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKYaaeWaaeaacqaIXaqmcqGHsislcqWGYbGCdaWgaaWcbaGaemyAaKMaem4CamNaem4Ba82aaSbaaWqaaiabdAfawbqabaaaleqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqWHXaqmdaWgaaadbaGaemyAaKMaem4CamNaem4Ba8gabeaaliabcIcaOiabdsha0jabcMcaPaaaaaa@5CA7@ for very sick cases at home,

β‴H,as,ai(t)=β″V,as,ai(t)(1−risoH)1iso(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGeamaaBaaaleaacqWGibascqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaagaWaaSbaaSqaaiabdAfawjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKYaaeWaaeaacqaIXaqmcqGHsislcqWGYbGCdaWgaaWcbaGaemyAaKMaem4CamNaem4Ba82aaSbaaWqaaiabdIeaibqabaaaleqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqWHXaqmdaWgaaadbaGaemyAaKMaem4CamNaem4Ba8gabeaaliabcIcaOiabdsha0jabcMcaPaaaaaa@5C6F@ for hospitalized very sick cases,

where 1iso (t) is a function which indicates when mass gathering events are possible or when they are closed:

1iso(t)={1⥄⥄⥄while isolation measures are performed0⥄⥄⥄while no isolation measures are performed.

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@AA0A@

The contact rates of cases in the late incubation period and that of asymptomatic cases remain unchanged:

β‴L,as,ai(t)=β″L,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGeamaaBaaaleaacqWGmbatcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaagaWaaSbaaSqaaiabdYeamjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKcaaa@4930@ for infected individuals in the late incubation period,

β‴M,as,ai(t)=β″A,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFYoGygaGeamaaBaaaleaacqWGnbqtcqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaagaWaaSbaaSqaaiabdgeabjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKcaaa@491C@ for asymptomatic cases.

Course of contagiousness

To allow for a contagiousness which changes over the course of disease, we multiply each contact rate with a weighting factor pk=xk−1/∑i=0m−1xi

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCdaWgaaWcbaGaem4AaSgabeaakiabg2da9maalyaabaGaemiEaG3aaWbaaSqabeaacqWGRbWAcqGHsislcqaIXaqmaaaakeaadaaeWbqaaiabdIha4naaCaaaleqabaGaemyAaKgaaaqaaiabdMgaPjabg2da9iabicdaWaqaaiabd2gaTjabgkHiTiabigdaXaqdcqGHris5aaaaaaa@4179@ whereby k is the stage of contagiousness. This leads to the following contact rates:

βAk,as,ai(t)=β‴A,as,ai(t)pk

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyqae0aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaasaWaaSbaaSqaaiabdgeabjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKIaemiCaa3aaSbaaSqaaiabdUgaRbqabaaaaa@4D82@ for asymptomatic cases in stage k,

βMk,as,ai(t)=β‴M,as,ai(t)pk

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaasaWaaSbaaSqaaiabd2eanjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKIaemiCaa3aaSbaaSqaaiabdUgaRbqabaaaaa@4DB2@ for moderately sick cases in stage k,

βVk,as,ai(t)=β‴V,as,ai(t)pk

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemOvay1aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaasaWaaSbaaSqaaiabdAfawjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKIaemiCaa3aaSbaaSqaaiabdUgaRbqabaaaaa@4DD6@ for very sick cases in stage k,

βHk,as,ai(t)=β‴H,as,ai(t)pk

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemisaG0aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaasaWaaSbaaSqaaiabdIeaijabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKIaemiCaa3aaSbaaSqaaiabdUgaRbqabaaaaa@4D9E@ for hospitalized cases in stage k.

For x = 1, contagiousness is equally high in all stages; for x = 0, only the first stage is contagious; for 0 We make the simplifying assumption that contagiousness does not change during the late incubation period

βLk,as,ai(t)=β‴L,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemitaW0aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPiabg2da9iqb=j7aIzaasaWaaSbaaSqaaiabdYeamjabcYcaSiabdggaHnaaBaaameaacqWGZbWCaeqaaSGaeiilaWIaemyyae2aaSbaaWqaaiabdMgaPbqabaaaleqaaOGaeiikaGIaemiDaqNaeiykaKcaaa@4ABA@ for cases in stage k = n - l,..,n of the incubation period.

Force of infection

To calculate the force of infection λas

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF7oaBdaWgaaWcbaGaemyyae2aaSbaaWqaaiabdohaZbqabaaaleqaaaaa@3185@ to which susceptible individuals of age as are exposed at time t, we have to first calculate the product of the number of contagious individuals with the corresponding contact rates and then to sum up these products over all ages ai, all risk categories r, all courses of the disease and all stages. Assuming that the contagiousness of cases who have received antiviral treatment is reduced by the factor (1 - fC), the force of infection is given by

λas(t)=∑ai(∑r∑k=n−l+1nβLk,as,ai(t)Ek,ai,r+∑k=1m(βAk,as,ai(t)Ak,ai+βMk,as,ai(t)⥄Mk,ai+βVk,as,ai(t)(Vk,ai+Wk,ai,U+(1−fI)Wk,ai,T+Xk,ai)+βHk,as,ai(t)(Hk,ai,U+(1−fI)Hk,ai,T)))

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@FCF8@

Differential equations for various model output

Cumulative number of deaths

D˙=∑a∑k=1m(τa,U(Xa+Hk,a,U)+τaHk,a,T)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGebargaGaaiabg2da9maaqafabaWaaabCaeaadaqadaqaaGGaciab=r8a0naaBaaaleaacqWGHbqycqGGSaalcqWGvbqvaeqaaOWaaeWaaeaacqWGybawdaWgaaWcbaGaemyyaegabeaakiabgUcaRiabdIeainaaBaaaleaacqWGRbWAcqGGSaalcqWGHbqycqGGSaalcqWGvbqvaeqaaaGccaGLOaGaayzkaaGaey4kaSIae8hXdq3aaSbaaSqaaiabdggaHbqabaGccqWGibasdaWgaaWcbaGaem4AaSMaeiilaWIaemyyaeMaeiilaWIaemivaqfabeaaaOGaayjkaiaawMcaaaWcbaGaem4AaSMaeyypa0JaeGymaedabaGaemyBa0ganiabggHiLdaaleaacqWGHbqyaeqaniabggHiLdaaaa@573B@

Convalescent (but non-contagious) cases

C˙1,a=γV,a,U(Vm,a+Wm,a,U+Xm,a+Hm,a,U)+γV,a,T(Vm,a,T+Hm,a,T)−ρC1,aC˙k,a=ρ(Ck−1,a+Ck,a)for k=2,...,j

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@98F1@

Immune and fully recovered individuals

I˙=∑a(ρCj,a+γAAm,a+γMMm,a)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGjbqsgaGaaiabg2da9maaqafabaWaaeWaaeaaiiGacqWFbpGCcqWGdbWqdaWgaaWcbaGaemOAaOMaeiilaWIaemyyaegabeaakiabgUcaRiab=n7aNnaaBaaaleaacqWGbbqqaeqaaOGaemyqae0aaSbaaSqaaiabd2gaTjabcYcaSiabdggaHbqabaGccqGHRaWkcqWFZoWzdaWgaaWcbaGaemyta0eabeaakiabd2eannaaBaaaleaacqWGTbqBcqGGSaalcqWGHbqyaeqaaaGccaGLOaGaayzkaaaaleaacqWGHbqyaeqaniabggHiLdaaaa@4BB9@

Number of people who are unable to work because of influenza

U˙=∑aW(∑rδEn,aW,r(caW,r(V)+caW,r(X))−τaW∑k=1m(Xk,aW+Hk,aW,U+Hk,aW,T)−ρCj,aW)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGvbqvgaGaaiabg2da9maaqafabaWaaeWaaeaadaaeqbqaaGGaciab=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@889B@

where aW denote all age classes of working adults (to avoid infinite contributions to the work loss, the decision was made that cases who die from influenza do not contribute any further to the total work loss).

Cumulative doses of antiviral treatment

T˙=α∑k=1ma,T∑a(fV(t)Vk,a+fX(t)Xk,a)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGubavgaGaaiabg2da9GGaciab=f7aHnaaqahabaWaaabuaeaacqGGOaakcqWGMbGzdaWgaaWcbaGaemOvayfabeaakiabcIcaOiabdsha0jabcMcaPiabdAfawnaaBaaaleaacqWGRbWAcqGGSaalcqWGHbqyaeqaaOGaey4kaSIaemOzay2aaSbaaSqaaiabdIfaybqabaGccqGGOaakcqWG0baDcqGGPaqkcqWGybawdaWgaaWcbaGaem4AaSMaeiilaWIaemyyaegabeaakiabcMcaPaWcbaGaemyyaegabeqdcqGHris5aaWcbaGaem4AaSMaeyypa0JaeGymaedabaGaemyBa02aaSbaaWqaaiabdggaHjabcYcaSiabdsfaubqabaaaniabggHiLdaaaa@56CF@

Initial values

Using the user-specified numbers of people Na in the age classes and the fractions Fa of people under high risk within each age class (Table 2), we obtain the initial population sizes according to age and risk class: Na,r1

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemyyaeMaeiilaWIaemOCai3aaSbaaWqaaiabigdaXaqabaaaleqaaaaa@32BD@ (0) = Na (1 - Fa) and Na,r2

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaemyyaeMaeiilaWIaemOCai3aaSbaaWqaaiabikdaYaqabaaaleqaaaaa@32BF@ (0) = NaFa. The total population is, therefore, given by N(0)=∑a∑rNa,r(0)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGobGtcqGGOaakcqaIWaamcqGGPaqkcqGH9aqpdaaeqbqaamaaqafabaGaemOta40aaSbaaSqaaiabdggaHjabcYcaSiabdkhaYbqabaGccqGGOaakcqaIWaamcqGGPaqkaSqaaiabdkhaYbqab0GaeyyeIuoaaSqaaiabdggaHbqab0GaeyyeIuoaaaa@4008@.

At time t = 0, one infection is introduced into an otherwise fully susceptible population. To avoid biasing the simulation one way or the other, the initial infection is distributed over all classes, weighted by the probability that an individual in one class acquires the infection (i.e. by the component of the eigenvector e→=(ea)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWHLbqzgaWcaiabg2da9iabcIcaOiabhwgaLnaaBaaaleaacqWGHbqyaeqaaOGaeiykaKcaaa@33A5@ of the next generation matrix):

Sa,r(0)=Na,r(0)−{(1−Fr)ea/∑aieaiif r=r1 (low risk group)Frea/∑aieaiif r=r2 (high risk group)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9C27@

Ek,a,r(0)={(1−Fr)Frea/∑aieaiif r=r1 (low risk group) and k=1Frea/∑aieaiif r=r2 (high risk group) and k=10if k>1

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B1F9@

∀k=1m

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHaiIidaqhaaWcbaGaem4AaSMaeyypa0JaeGymaedabaGaemyBa0gaaaaa@3261@Ak, a (0) = Mk, a (0) = Vk, a (0) = Wk, a, U (0) = Wk, a, T (0) = Xk, a (0) = Hk, a, U (0) = Hk, a, T (0) = 0

∀k=1j

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHaiIidaqhaaWcbaGaem4AaSMaeyypa0JaeGymaedabaGaemOAaOgaaaaa@325B@Ck, a (0) = 0, D (0) = I (0) = U (0) = T (0) = 0.

Using these initial values, the set of differential equations is solved numerically with a Runge-Kutta method with step-size control.

Transmission variables

Sa, r number of susceptible individuals

Ek, a, r number of incubating individuals (stage k); the last two stages are contagious

Ak, a number of asymptomatic individuals (stage k)

Mk, a number of moderately sick individuals (stage k)

Vk, a number of very sick individuals who have not yet seen a doctor (stage k)

Wk, a, T number of treated very sick individuals (withdrawn to home; stage k)

Wk, a, U number of untreated very sick individuals (withdrawn to home; stage k)

Xk, a number of extremely sick individuals who have not seen a doctor (stage k)

Hk, a, T number of hospitalized and treated individuals (stage k)

Hk, a, U number of hospitalized but untreated individuals (stage k)

Output variables

Ck, a number of convalescent (non-contagious) cases (stage k)

I number of fully recovered and immune cases

D number of people who die of influenza

U number of people who are unable to work because of influenza

T cumulative number of antiviral treatment doses used

Parameters concerning the demography

Na total population size by age class a, whereby a = a1 denotes children, a = a2 denotes adults of working age and a = a2 denotes elderly, respectively.

Fa fraction of the population in age class a which is under high risk from this, Na, r is calculated such that Na, r = Fara

Kas,ai

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGlbWsdaWgaaWcbaGaemyyae2aaSbaaWqaaiabdohaZbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaemyAaKgabeaaaSqabaaaaa@34A7@ the contact matrix gives the weekly number of contacts between an individual of age class ai with individuals of age class as. From this, the contact rates βLk,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemitaW0aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPaaa@3BF5@, βAk,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyqae0aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPaaa@3BDF@, βMk,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPaaa@3BF7@ and βVk,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemOvay1aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPaaa@3C09@ are calculated as explained above

Number of stages

n number of stages used to model the latent period

l number of stages used to model the early infectious period

m number of stages used to model the (symptomatic) infectious period

j number of stages used to model convalescence

Sojourn times

DE average duration of the incubation period;

δ is calculated such that δ = n/DE

the last l stages are used as early infectious period

(average duration: DL = DEl/n)

DD average time after onset when a severe case seeks medical help;

α is calculated such that α = 1/DD

DA, a average infectious duration for asymptomatic cases

γA, a is calculated such that γA, a = m/DA, a

DM, a average infectious duration of moderately sick cases

γM, a is calculated such that γM, a = m/DM, a

DV, a average duration of infectivity of untreated very or extremely sick cases;

γV, a, U is calculated such that γV, a, U = m/DV, a

DC average duration of convalescence;

ρ is calculated such that ρ = j/DC

Course of disease

ca, r (A) fraction of asymptomatic infections (given age a and risk r)

sa, r fraction of severe cases among symptomatic ones

ha, r fraction of severe cases who need hospitalization (unless treated) the fraction of infected cases who

- develops moderate disease is ca, r (M) = (1 - sa, r)(1 - ca, r (M))

- becomes bed-ridden at home is ca, r (V) = sa, r (1 - ha, r)(1 - ca, r (M))

- become extremely severe cases is ca, r (X) = sa, rha, r (1 - ca, r (M))

da fraction of untreated extremely severe cases who die;

from this, τa is chosen such that da=τaτa+γS,a,U∑k=0m−1(τaτa+γS,a,U)k

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazdaWgaaWcbaGaemyyaegabeaakiabg2da9maalaaabaacciGae8hXdq3aaSbaaSqaaiabdggaHbqabaaakeaacqWFepaDdaWgaaWcbaGaemyyaegabeaakiabgUcaRiab=n7aNnaaBaaaleaacqWGtbWucqGGSaalcqWGHbqycqGGSaalcqWGvbqvaeqaaaaakmaaqahabaWaaeWaaeaadaWcaaqaaiab=r8a0naaBaaaleaacqWGHbqyaeqaaaGcbaGae8hXdq3aaSbaaSqaaiabdggaHbqabaGccqGHRaWkcqaHZoWzdaWgaaWcbaGaem4uamLaeiilaWIaemyyaeMaeiilaWIaemyvaufabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabdUgaRbaaaeaacqWGRbWAcqGH9aqpcqaIWaamaeaacqWGTbqBcqGHsislcqaIXaqma0GaeyyeIuoaaaa@59E6@

Parameters concerning the contagiousness of the infection

bL relative contagiousness of cases in the late incubation period

bA relative contagiousness of asymptomatic cases

bM relative contagiousness of moderately sick cases

bV relative contagiousness of severely sick cases

x50 parameter regulating the course of contagiousness

x50 = 1 only the first stage after onset of disease is contagious

0.5

x50 = 0.5 equal contagiousness during the whole course of disease

0

from this, x is calculated such that x50=∑i=0m/2xi−1/∑i=0mxi−1

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaWgaaWcbaGaeGynauJaeGimaadabeaakiabg2da9maalyaabaWaaabCaeaacqWG4baEdaahaaWcbeqaaiabdMgaPjabgkHiTiabigdaXaaaaeaacqWGPbqAcqGH9aqpcqaIWaamaeaacqWGTbqBcqGGVaWlcqaIYaGma0GaeyyeIuoaaOqaamaaqahabaGaemiEaG3aaWbaaSqabeaacqWGPbqAcqGHsislcqaIXaqmaaaabaGaemyAaKMaeyypa0JaeGimaadabaGaemyBa0ganiabggHiLdaaaaaa@4ACD@ if m is an even number or x50=(∑i=0(m−1)/2xi−1+x(m−1)/2+12)/∑i=0mxi−1

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5BEC@ if m is an odd number, respectively

R0 basic reproduction number; the contact rates βLk,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemitaW0aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPaaa@3BF5@, βAk,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyqae0aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPaaa@3BDF@, βMk,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemyta00aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPaaa@3BF7@ and βVk,as,ai(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFYoGydaWgaaWcbaGaemOvay1aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGHbqydaWgaaadbaGaem4CamhabeaaliabcYcaSiabdggaHnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcIcaOiabdsha0jabcMcaPaaa@3C09@ are calculated from R0 and from the contagiousness factors as explained above

λa (t) force of infection for susceptible individuals of age a at time t (see calculation above)

Parameters concerning contact reduction

risoM

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaWgaaWcbaGaemyAaKMaem4CamNaem4Ba82aaSbaaWqaaiabd2eanbqabaaaleqaaaaa@33D1@ fraction of contacts of moderately sick patients that are prevented by partial isolation

risoV

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaWgaaWcbaGaemyAaKMaem4CamNaem4Ba82aaSbaaWqaaiabdAfawbqabaaaleqaaaaa@33E3@ fraction of contacts of very sick patients that are prevented by partial isolation

risoH

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaWgaaWcbaGaemyAaKMaem4CamNaem4Ba82aaSbaaWqaaiabdIeaibqabaaaleqaaaaa@33C7@ fraction of contacts of hospitalized patients that are prevented by partial isolation

rgen general fraction of contacts that are prevented at time t

rmass fraction of contacts among (healthy) adults that are prevented by cancelling events of mass gatherings at time t

rsch fraction of contacts among (healthy) children of the same age class that occurs in day care centres or schools

FHC factor by which the contacts between adults and severely sick children increase because of child health care

FCC factor by which the contacts between adults and children increase when children are taken care off at home because schools are closed

Parameters concerning antiviral treatment

Tmax available number of antiviral treatment doses

DT time after onset until when antiviral treatment can still be given; the latest infectious stage ma, T during which treatment can be given, is chosen such that ma, T/γV, a, U ≤ DT ≤ (ma, T + 1)/γV, a, U

fV fraction of severe cases eligible to receive antiviral treatment; treatment will be given only in the user-specified time window and only as long as supplies last:

fV(t)={fVif T(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7404@

fX fraction of extremely severe cases eligible to receive antiviral treatment; treatment will be given only in the user-specified time window and only as long as supplies last:

fX(t)={fXif T(t)

MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@740C@

fD fraction by which the duration of infectiousness is reduced by antivirals; γV, a, T is calculated from this such that γV, a, T = m/((1 - fD)DV, a)

fI fraction by which the infectiousness of treated cases is reduced by antivirals

fH fraction of hospitalizations prevented by antiviral treatment

Pre-publication history

The pre-publication history for this paper can be accessed here: