On SIR-type epidemiological models and population heterogeneity eects Silke Klemm12and Lucrezia Ravera34

2025-05-02 0 0 1.12MB 16 页 10玖币

侵权投诉

On SIR-type epidemiological models and population

heterogeneity eﬀects

Silke Klemm1,2∗and Lucrezia Ravera3,4†

1Dipartimento di Fisica, Universit`a di Milano, Via Celoria 16, 20133 Milano, Italy

2INFN, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy

3DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

4INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

October 21, 2022

Abstract

In this paper we elaborate on homogeneous and heterogeneous SIR-type epidemiological

models. We ﬁnd an unexpected correspondence between the epidemic trajectory of a transmis-

sible disease in a homogeneous SIR-type model and radial null geodesics in the Schwarzschild

spacetime. We also discuss modeling of population heterogeneity eﬀects by considering both a

one- and two-parameter gamma-distributed function for the initial susceptibility distribution,

and deriving the associated herd immunity threshold. We furthermore describe how mitigation

measures can be taken into account by model ﬁtting.

∗silke.klemm@mi.infn.it

†lucrezia.ravera@polito.it

arXiv:2210.11342v1 [q-bio.PE] 20 Oct 2022

Contents

1 Introduction 1

2 Homogeneous SIR-type models 2

2.1 Epidemic trajectory as radial null geodesics in Schwarzschild spacetime . . . . . . . 5

3 Modeling population heterogeneity eﬀects 6

3.1 HIT for the case of a one-parameter gamma distribution . . . . . . . . . . . . . . . . 7

3.1.1 A possible way to take into account mitigation measures in model ﬁtting . . 10

3.2 HIT for the case of a two-parameter gamma distribution . . . . . . . . . . . . . . . . 10

4 Final remarks 13

1 Introduction

The SARS-CoV-2 pandemic led, in many countries, to lockdown measures aiming to control

and limit the spreading of the virus. A key role when facing global events of this type is played

by mathematical modeling of infectious diseases, which allows direct validation with real data.

This consequently permits to evaluate the eﬀectiveness of control and prevention strategies, giving

support to public health.

In this context, Susceptible-Infected-Removed (SIR) models of epidemics (see e.g. [1,2]) capture

key features of a spreading epidemic as a mean ﬁeld theory based on pair-wise interactions between

infected and susceptible individuals, without aiming to describe speciﬁc details. In particular, in the

presence of Iinfected individuals in a population of Nindividuals, the infection can be transmitted

to susceptible individuals S. They stay infectious during an average time γ−1, after which they

no longer contribute to infections. The fraction of immune individuals in the population beyond

which the epidemic can no longer grow deﬁnes the herd immunity threshold (HIT).

Simple SIR models commonly assume the population to be homogeneous; each individual has

the same probability of being infected by the disease. However, in order to take into account that

the infection probability actually depends on age, sex, connections with other individuals, etc., SIR

models for heterogeneous populations have been considered [3,5–7]. In these models, a parameter,

usually denoted by α, is commonly introduced to describe population heterogeneity and, hence,

variation in susceptibility of individuals.

Studying the transmission of the virus SARS-CoV-2, in [3] it was shown that the percentage of

a homogeneous population to be immune given some value for R0(which is the basic reproduction

number, namely the average number of new infected generated by an infected individual at the early

epidemic stage) noticeably drops if the population is considered to be highly heterogeneous. More

speciﬁcally, while herd immunity is expected to require 60-75 percent of a homogeneous population

to be immune given an R0(that is the basic reproduction number) between 2.5 and 4, these per-

centages drop to the 10-20 percent range for the coeﬃcients of variation in susceptibility considered

in [3] between 2 and 4. In particular, it was shown that individual variation in susceptibility or

exposure (connectivity) accelerates the acquisition of immunity in populations due to selection by

the force of infection. More susceptible and more connected individuals have a higher propensity

to be infected and thus are likely to become immune earlier. Due to this selective immunization,

heterogeneous populations require less infections to cross their HITs than homogeneous (or not

suﬃciently heterogeneous) models would suggest. In [3] the initial susceptibility was considered to

be gamma-distributed, with a one-parameter gamma distribution. Besides, the case of a lognormal

distribution was treated numerically. The gamma distribution was also considered in [4] to model

the ﬁrst-wave COVID-19 daily cases, and it was proven, in this context, to provide better results

than the Gaussian, Weibull (and Gumbel) distributions.

Taking into account heterogeneity eﬀects has proven to be relevant also in the spread of smallpox

(cf. [7]), where homogeneous models are not capable to explain the data, as well as for tuberculosis

and malaria (see, e.g., [3] and references therein).

In this work we discuss modeling of population heterogeneity eﬀects by considering both a

one-parameter gamma-distributed function and a two-parameter one for the initial susceptibility

distribution, deriving the associated HIT. The latter is computed analytically in both cases. We

also describe a possible way to take into account mitigation measures when performing model ﬁtting

in the case of the one-parameter initial gamma distribution, while the two-parameter initial gamma

distribution appears to automatically accommodate this external action on diseases spread. On

the other hand, regarding homogeneous SIR models, we present an intriguing feature of a simple

model of this type, which paves the way to future analytically tractable studies of epidemiological

models.

The remainder of this paper is structured as follows: In Section 2, we review homogeneous SIR-

type models of epidemics and, in Section 2.1, we present a correspondence between the epidemic

trajectory in a homogeneous SIR model and radial null geodesics in the Schwarzschild spacetime.

Subsequently, in Section 3, we discuss modeling of population heterogeneity eﬀects to capture the

fact that the probability of being infected is not the same for all individuals. Section 4is devoted

to ﬁnal remarks and possible future developments of our analysis.

2 Homogeneous SIR-type models

In an SIR-type model [1], the population is divided into susceptible, infected and recovered

individuals, whose numbers are denoted respectively by S,I, and R. Their dynamics is governed

by the equations ˙

S=−f(I, S),˙

I=f(I, S)−g(I),˙

R=g(I).(1)

Here f(I, S) denotes the infection force, i.e., the rate at which susceptible persons acquire the

infectious disease, while g(I) is some function to be speciﬁed below. The upper dot symbol denotes

the time derivative. From (1) one obtains the conservation law

S+˙

I+˙

R= 0 ⇒S+I+R= const. = N , (2)

with Nthe total number of individuals in the population. A common choice is f(I, S) = βIS,

g(I) = γI, where βis the transmission (or infection) rate (per capita),1and γdenotes the rate of

1The infection rate βcan in general depend on time t; this time dependence could correspond to seasonal changes

or mitigation measures [8–10].

recovery. It is related to the average recovery time Dby D= 1/γ. We have thus2

S=−βIS , ˙

I=βIS −γI , ˙

R=γI . (3)

This implies dI

dS =−1 + γ

βS ,(4)

which can be integrated to give the epidemic trajectory

I−I0=S0−S+γ

βln S

,(5)

with I0=I(t= 0) and S0=S(t= 0). In order to obtain the early growth of the epidemic, one

linearizes (3) around S=S0≈Nand I≈0, i.e., sets

S=N−δS , I =δI , δS, δI N . (6)

This leads to the exponential law

δI =I0eγ(R0−1)t,(7)

where

R0=Nβ

γ(8)

is the basic reproduction number. It denotes the average number of new infections generated by

an infected individual (at the early epidemic stage).

The function I(S) has a maximum at S=Smax =γ/β. At this peak, a fraction Smax/N = 1/R0

of individuals remains susceptible. The cumulative number of infections C=N−Sat the maximum

of Ithus obeys Cmax

N= 1 −1

.(9)

This is the well-known formula for the herd immunity level (or herd immunity threshold, HIT),

i.e., the fraction of immune individuals in the population beyond which the epidemic can no longer

grow. Here we are not considering mitigation measures, nor reinfections. Hence, in particular, the

threshold to reach herd immunity is estimated by considering natural infections without restrictions

(lockdown, social distancing, etc.) and without taking into account possible vaccinations. The plot

in Figure 1displays the herd immunity level (9) as a function of R0.

When the epidemic stops we have I= 0. Using (5), it is easy to show that the number of

susceptible individuals left over at the end of an epidemic is given by

S=−S0

W0−Reexp −Re1 + I0

S0,(10)

where

Re=S0β/γ =S0R0/N (11)

2In this work, we multiply the quantity βwith the constant factor Nwith respect to the one deﬁned, e.g., in [7],

that is β→Nβ.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

OnSIR-typeepidemiologicalmodelsandpopulationheterogeneityeectsSilkeKlemm1;2*andLucreziaRavera3;41DipartimentodiFisica,UniversitadiMilano,ViaCeloria16,20133Milano,Italy2INFN,SezionediMilano,ViaCeloria16,20133Milano,Italy3DISAT,PolitecnicodiTorino,CorsoDucadegliAbruzzi24,10129Torino,Italy4INFN,Sezi...

展开>> 收起<<

On SIR-type epidemiological models and population heterogeneity eects Silke Klemm12and Lucrezia Ravera34.pdf

共16页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

On SIR-type epidemiological models and population heterogeneity eects Silke Klemm12and Lucrezia Ravera34

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: