MATH0011 Lecture 9

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MATH0011Numbers and Patterns in

Nature and Life

Lecture 9Infectious disease modeling II

http://147.8.101.93/MATH0011/

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Review of SIR modelDivide the total population N into three groups:

St + It + Rt = N

Infective

It

Recovered or immune

Rt

γ ItSusceptible

St

αSt It

St+1 = St − α St It

It+1 = It + α St It − γ It

Rt+1 = Rt + γ It

The SIR model does not fit all infectious diseases.

However, for certain diseases, some slight variations of the SIR model may fit very well.

We shall consider two such variations ─the SI model and the SIS model ─ both of which do not have the removed class.

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Variations of the SIR modelThe SI model assumes that St + It = N.

It assumes that there is no removed class, and that once an individual becomes infective, she/he will stay in the infective class for the duration of the model.

E.g., the spread of head lice in a human population with no means of effective treatment would fit the SI model.

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The SI model

Susceptible

St

Infective

It

αSt It

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The equations for the SI model are

In this model, St is always non-increasing, and Itis always non-decreasing.

Also, since St + It = N is a constant, knowing one of St or It will give the value of the other.

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The SI model (cont.)

St+1 = St − α St It

It+1 = It + α St It

Another variation is the SIS model.

It has the susceptible class and the infective class. But when infectives recovered, they are at risk again for contracting the disease. Therefore there is no removed class.

Examples of such diseases are syphilis and gonorrhea, which can be treated with antibiotics. But a patient, once cured, may become re-infected.

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The SIS model

Susceptible

St

Infective

It

αSt It

γ It

The equations for the SI model are

The SIS model can lead to a constant, yet nonzero number of infectives in a population. When this happens we say that the disease is endemic.

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The SIS model (cont.)

St+1 = St − α St It + γ It

It+1 = It + α St It − γ It

Infectious disease models are often formulated a bit differently from we did before. Quite often, instead of absolute numbers, proportions of the population are used.

This approach is useful when the precise size of the population may not be known, or does not matter.

This formulation allows us to replace the transmission coefficient α with a new parameter that has a more meaningful biological interpretation.

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The sir model

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Let us still consider the SIR model. Define

s = S∕N , i = I∕N , r = R∕N.

Here s , i , r are the proportions of the S , I , Rclasses within the population.

Hence st + it + rt = 1.

We continue to assume there are no births or immigration, etc., so that the total population size N remains constant, even though N will not appear in our equations.

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The sir modelDefine Δs = st +1 − st , which is the change in proportion of the S class. Define Δi , Δr similarly.

Through a similar thought process as used in the SIR model, the s i r equations are obtained:

where β is the contact rate. Exercise: show that β = α N .

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The sir model

Δs = − β st itΔi = β st it − γ itΔr = β it

Note that, if we introduce a specific total population size N, we can recover SIR data from the sir model simply by multiplying N to the s i rformulas. For example, because

ΔS = St+1 − St = N st+1 − N st = N Δs

the net change in number of susceptible is:

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− (β it) St=− (β it) (st N)=− β st it N=(Δs) N=ΔS

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By the term contact we mean an interaction between individuals that is sufficient for disease transmission.

Note that ΔS equals to the number of susceptible infected by the infectives in a single time step, and infection occurs only when a susceptible have contact with an infective, not with a healthy or immune individual susceptible.

Therefore, the total number of contacts the susceptibles have with infectives during a single time step should be equal to ΔS .

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Since there are St susceptibles, the average number of contacts a susceptible has with infectives during a single time step should be

ΔS∕St = － β itbecause ΔS = − (β it) St .

Therefore βit represents the average number of contacts a susceptible has with infectives during a single time step.

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Because it denotes the fraction of the population in the infective class, we can interpret β as the average number of contacts that an individual has during a single time step.

Therefore β measures contacts between anyone, whether infective or not, that would have caused an infection if one of the individuals was infective and the other susceptible.

Meaning of the contact rate β

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The value of β depends on the particulars of the disease under study.

For example, because chickenpox is highly contagious, it seems plausible that an elementary school child might have contact, sufficient to spread chickenpox, with four other children throughout the course of a day. In this case, β = 4 and Δt = 1 day.

For a less contagious disease, β might be a small number such as 0.02.

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Another important value is the contact number σ , the average number of contacts of a typical infective during the entire infectious period.

This is a “per-infective” measurement. For the s i rmodel above, we multiply the contact rate β by the mean infectious period 1∕γ to find

σ = β∕γ.

The contact number σ is closely related to the basic reproduction number P0 .

Exercise: work out the relationship between P0 and σ , and see why they are essentially the same as long as S0 is almost as big as N.

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Immunization

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Vaccinations, when available, are an attractive way to control disease dynamics. One goal of any immunization program is to achieve herd immunity, (i.e., to ensure that no epidemic can take place even if a few cases of the disease are present).

The s i r model is useful to help us to estimate what level of vaccination confers herd immunity for a particular disease in a particular society.

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As an example, consider a large population at risk for a disease modeled well by the s i r equations.

We would like to ensure that, regardless of the size i0 of the fraction of the population that was infective, it never increases. Thus, we want

Δi = β st it － γ it < 0 .

Since β st it － γ it =(β st － γ) it and it ≧ 0, this means we want

β st － γ < 0, or st < γ∕β = 1∕σ .

In other words, 1∕σ is the threshold value determining whether disease spreads or not.

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Hence, if the fraction of the population that is susceptible can be brought below 1/σ then an epidemic cannot occur.

To ensure herd immunity, the fraction of the population that must be vaccinated successfully must be at least 1－1/σ .

However, there are other social and economic factors that affect the planning of an vaccination policy. E.g., individuals receiving vaccinations are usually at some small risk of adverse effects. The US and UK stopped routine vaccinations for smallpox in 1971, as there were evidence showing that more people were dying from complications arising from vaccination than would have died from smallpox itself.

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The pattern of the SARS outbreak in Hong Kong was puzzling after a residential estate (Amoy Gardens) in Hong Kong was affected, with a huge number of patients infected by the virus causing SARS. In particular it appeared that underlying this highly focused outbreak there remained a more or less constant background infection level. This pattern is difficult to explained by the standard SIR epidemic model.

SARS outbreak in Hong Kong

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This pattern is difficult to explain with the standard SIR model, why?

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Figure 2: Daily new number of confirmed SARS cases from Hong Kong: hospital, community and the Amoy Gardens.

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Note that daily new number of confirmed SARS cases from the community is assumed to be ΔR= R t+1-R t.What should be the shape of the graph ofΔR= R t+1-R t in the SIR model?

It should be the same as that of It because

ΔR= γIt

Since the shape of the blue curve is very different from that of the I curve, SIR model is unlikely to explain the outbreak.

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Infectious It

Recovered / immune

Rt

Susceptible St

SEIR model - including latent period

Add one more class -- the exposed class which includes people who are infected but cannot transmit the disease to others.

tttt IrSSS −=+1

Exposed

Et

r b a

ttttt bEIrSEE −+=+1

tttt aIbEII −+=+1

ttt aIRR +=+124

)()( tItrSdtdS

−=

)()()( tbEtItrSdtdE

−=

)()()( taItbEtIdtdI

−+=

)(taIdtdR

=

Replace ΔS = by dS/dt, ΔE by dE/dt etc. We get a system of four ordinary differential equations describing the SEIR model:

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SEIR model -- including latent periodInfectious

It

Recovered / immune Rt

Susceptible St

Exposed

Et

r b a

I was not able to find suitable parameters, r, b and a so that the predictions from the SEIR model is compatible with the data.Try another model.

For given r, b and a, we can solve the systems of differential equations by using some mathematical software like Mathlab.

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Researcher Team Members

Patrick Ng T.W.Department of Mathematics, HKUGabriel TuriniciINRIA, Domaine de Voluceau, Rocquencourt, FranceAntoine DanchinGénétique des Génomes Bactériens,Institut Pasteur, Paris, France(Former director of HKU-Pasteur Institute)

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A Double Epidemic Model for the SARS Propagation

Published in BMC Infectious Diseases2003, 3:19 (10 September 2003)

Can be found online at:http://www.biomedcentral.com/1471-2334/3/19

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There are two epidemics, one is SARS caused by a coronavirus, call it virus A.

Hypothesis of the Double Epidemic Model for the SARS Propagation

Another epidemic (disease B) is caused by an unknown virus B.

Both epidemics would spread in parallel, and we assume that the epidemic caused by virus B which is rather innocuous, protects against SARS(so that naïve regions, not protected by the epidemic B can get SARS large outbreaks).

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Another epidemic (disease B), which may have appeared before SARS, is assumed to be extremely contagious because of the nature of the virus and of its relative innocuousness, could be propagated by contaminated food and soiled surfaces. It could be caused by some coronavirus, call it virus B. The most likely is that it would cause gastro-enteritis.The most likely origin of virus A is a more or less complicated mutation or recombination event from virus B.

Some suggestions for disease B

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The previous suggestions are based on a set of coronavirus mediated epidemics happened in Europe that affected pigs in the 1983-1985.It is known at that time a coronavirus and its variant caused a double epidemic when the mutated virus changed its original tropism from the small intestine to lung.

Motivation of a Double Epidemic Model for the SARS Propagation

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This in a way allowing the first one to provide some protection to part of the exposed population.D Rasschaert, M Duarte, H Laude: Porcine respiratory coronavirus differs from transmissible gastroenteritis virus by a few genomic deletions. J Gen Virol 1990, 71 ( Pt 11):2599-607.

Motivation of a Double Epidemic Model for the SARS Propagation

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The suggestions are also based on two facts:A) the high mutation and recombination

rate of coronaviruses.

SR Compton, SW Barthold, AL Smith: The cellular and molecular pathogenesis of coronaviruses. Lab Anim Sci 1993, 43:15-28.

A Double Epidemic Model for the SARS Propagation

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B) the observation that tissue tropismcan be changed by simple mutations .

BJ Haijema, H Volders, PJ Rottier: Switching species tropism: an effective way to manipulate the feline coronavirus genome. J Virol 2003, 77:4528-38.

A Double Epidemic Model for the SARS Propagation

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Assume that two groups of infected individuals are introduced into a large population. One group is infected by virus A.The other group is infected by virus B.Assume both diseases which, after recovery, confers immunity (which includes deaths: dead individuals are still counted). Assumed that catching disease B first will protect the individual from disease A.

A Double Epidemic SEIRP Model

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We divide the population into six groups:— Susceptible individuals, S(t)— Exposed individuals for virus A, E(t) — Infective individuals for virus A, I(t)— Recovered individuals for virus A, R(t)— Infective individuals for virus B, Ip(t)— Recovered individuals for virus B, Rp(t)

A Double Epidemic SEIRP Model

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The progress of individuals is schematically described by the following diagram.

r

b

apr p

a

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The system of ordinary differential equations describes the SEIRP model:

)()()()( tItSrtItrSdtdS

Pp−−= (1)

)()()( tbEtItrSdtdE

−= (2)

)()( taItbEdtdI

−= (3)

)(taIdtdR

= (4)

)()()( tIatItSrdt

dIPPPP

P −= (5)

)(tIadt

dRPP

P = (6)

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One should try to make the model as simple as possible because the more assumptions you put into the model, the easier you can “explain” the phenomena.Therefore we would like to keep the number of parameters as small as possible. Because of this, we did not include the exposed class for the unknown disease B. This also save us from searching the extra parameters.

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Meaning of some parameters

As the SIR model, we can interpret 1/b as the length of the latent period.

Similarly, we interpret 1/a as the length of the infectious period.Note that for an individual patient, it is difficult to know the length of the latent period and the length of the infectious period.

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IndividualIndividual’’s s disease disease state :state : LatentLatent InfectiousInfectious

Immune / Immune / RemovedRemoved

timetimeEpoch :Epoch :

Onset to admission time interval

Incubation periodttAA tBB tCC tDD tEE

Infection Infection occursoccurs

Latency to Latency to infectious infectious transitiontransition

Symptoms Symptoms appearappear

First First transmission transmission to another to another susceptiblesusceptible

Individual no Individual no longer longer

infectious to infectious to susceptiblessusceptibles(recovery or (recovery or

removal)removal)

Note : tD is constrained to lie in the interval (tB , tE), so tD > tC(as shown) and tD < tC are both possible.

SusceptibleSusceptible

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Meaning of some parameters

The incubation period (the time from first infection to the appearances of symptoms) plus the onset to admission interval is equal to the sum of the latent period and the infectious period and is therefore equal to 1/b + 1/a.

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Empirical StatisticsCA Donnelly, et al., Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong, The Lancet, 2003.The observed mean of the incubation period for SARS is 6.37 days. The observed mean of the time from onset to admission is about 3.75 days.Therefore, the estimated 1/a + 1/b has to be close to 6.37+3.75=10.12.

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Parameter Estimations

Since we do not know how many Hong Kong people are infected by virus B, we shall consider the following two scenarios.Case a: Assume Ip(0)=0.5 million, S(0)=6.8-0.5=6.3 million,E(0)=100,I(0)=50.

Case b: Assume Ip(0)=10, S(0)=6.8 million,E(0)=100,I(0)=50.

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We fit the model with the total number of confirmed cases from 17 March, 2003 to 10 May, 2003 (totally 55 days). The parameters are obtained by the grid search.The resulting curve for R fits very well

with the observed total number of confirmed cases of SARS from the community.

Parameter Estimations

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Figure 3: Number of SARS cases in Hong-Kong community (and the simulated case “a”) per three days.

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Figure 4: Number of SARS cases in Hong-Kong community (and the simulated case “b”) per three days.

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Parameter Estimations

Case a: Assume Ip(0)=0.5 million, S(0)=6.3 million,E(0)=100,I(0)=50.

r=10.19x10-8, r_p=7.079x10-8 .a=0.47,a_p=0.461,b=0.103.Estimated 1/a + 1/b = 11.83 (quite close to the observed 1/a+1/b= 10.12).

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Parameter Estimations

Case b: Assume Ip(0)=10, S(0)=6.8 million,E(0)=100,I(0)=50.

r=10.08x10-8, r_p=7.94x10-8.a=0.52,a_p=0.12,b=0.105.Estimated 1/a + 1/b = 11.44 (quite close to the observed 1/a+1/b= 10.12).

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Basic reproductive factor

P0 is the number of secondary infections produced by one primary infection in a whole susceptible population.Case a: P0=1.37.Case b: P0=1.32.

Like what we do in the SIR model,we define the basic reproductive factor P0 as

P0=r S(0)/a.

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ConclusionOur hypothesis assumes the existence of a mild epidemic protects against SARS. To test this hypothesis directly, viral studies of the people in Hong Kong should be carried out to determine if the unknown virus actually exists.

Before carrying out the costly tests, a mathematical model based on the hypothesis may be built first to see if the observations can be explained by the model.The estimations from the model are compatible with the observations and therefore this gives certain support to the hypothesis.

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ConclusionIn 2004, the research group leaded by Prof. Kwok Yung Yuen discovered a new coronavirus (named as CoV-HKU1) from a 71-year-old patient with pneumonia.

The research group then screened among the 400 non-SARS patients with respiratory illness during the SARS period and found out that one sample also contains this new coronavirus which is named as

Coronavirus HKU1Characterization and Complete Genome Sequence of a Novel Coronavirus, Coronavirus HKU1, from Patients with Pneumonia , J. Virol. 2005. 79: 884-895.