Theoretical Biology and Medical

26 Jun Theoretical Biology and Medical

ommentary

Formal kinetics of H1N1 epidemic

Konstantin G Gurevich

Open Access

Address: UNESCO Chair in Healthy Life for Sustainable Development, Moscow State University of Medicine and Dentistry (MSDMU), Delegatskaya Street 20/1, 127473, Moscow, Russia

Email: Konstantin G Gurevich – [email protected].ru

Published: 15 September 2009

Received: 20 July 2009

Theoretical Biology and Medical Modelling2009,6:23

doi:10.1186/1742-4682-6-23

Accepted: 15 September 2009

This article is available from: http://www.tbiomed.com/content/6/1/23

© 2009 Gurevich; licensee BioMed Central Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Background:The formal kinetics of the H1N1 epidemic seems to take the form of an exponential curve. There is a good correlation between this theoretical model and epidemiological data on the number of H1N1-infected people. But this formal model leads to paradoxes about the dates when everyone becomes infected: in Mexico this will happen after one year, then in the rest of the world.

Further implications of the formal model:The general limitations of this formal kineticsmodel are discussed. More detailed modeling is examined and the implications are examined in the light of currently available data. The evidence indicates that not more than 10% of the population is initially resistant to the H1N1 virus.

Conclusion:We are probably only at the initial stage of development of the H1N1 epidemic.Increasing the number of H1N1-resistant people in future (e.g. due to vaccination) may influence the dynamics of epidemic development. At present, the development of the epidemic depends only on the number of people in the population who are initially resistant to the virus.

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Background

The first cases of swine influenza virus A (H1N1) were reported on 9 April 2009 in Mexico and 17 April 2009 in the USA [1]. For some experts, the H1N1 epidemic seems to be a potential global disaster [2]. Every 2-3 days, WHO publishes updated information [3] about the number of H1N1-infected people all over the world.

Formal kinetics model

We took the number of H1N1-infected people in the countries showing the greatest spread from “zero time” (9 April) up to the most recent WHO update (6 Jul 2009). In half-logarithmic coordinates the number of infected peo-ple seems to be a linear function of time (Figure 1). From the formal kinetics point of view, this means that the number of H1N1-infected people can be described by the following function:

N= aexp(bt+ c)

(1)

where N is number of infected people, a, b and c are con-stants, and t is time.

The parameters of formal model (1) were calculated and there was a very good correlation coefficient between the theoretical model and real epidemiological data for differ-ent countries and for the world (Table 1).

Coefficient b is proportional to the velocity of the process:

dN

= bN

(2)

dt

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Theoretical Biology and Medical Modelling2009,6:23

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Figure 1

The number of infected people in different countries.

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i.e. the formal kinetics model of H1N1 epidemic develop-ment proposes that the velocity of the epidemic is directly proportional to the number of infected people. This implies that the H1N1 epidemic is only in its initial period and the number of H1N1-resistant people has not so far limited the epidemiological process.

Coefficient b seems to not to differ much among coun-tries: its minimum is in the USA and its maximum in China, but the difference between minimum and maxi-mum is only four-fold.

Infection of the total population

From the formal kinetics model (1), the “last” time (tlast), the time when all people in the country or in the world

will be infected, can be easily calculated:

http://www.tbiomed.com/content/6/1/23

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Figure 2

Model (4) numerical solution for the World. b”=b’/10.

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where N0 is the population size.

The calculated tlast is given in Table 1. It characterizes the “pessimistic” variant of H1N1 epidemic development. According to the formal model, 19 Feb 2010 will be the day when all people in the world will be H1N1-infected. But in Mexico, all people will be infected only on 14 Jul 2011. This is the paradoxical result obtained from the for-mal kinetics model as it stands.

As a final comment we note that the formal model (2) does not take many relevant factors into account, i.e. active or passive immunity to H1N1 infection, different population densities in different countries and so on. In other words, this model has serious limitations as a prac-tical approach.

Avoiding the paradox

1

N

0

In a real situation in the human population we have at

t last

=

ln

− c

(3)

least three types of people: infected (N), sensitive (M) and

b

a

Table 1: Calculated values for model (1) for different countries

Country

Number of citizens

a

b

c

tlast

Correlation coefficient between

[5], million people

model and epidemiological data, %

Canada

30

0.9249

0.06121

3.961

12.11.2009

95.3

China

1314.5

0.7458

0.1252

-2.8695

18.10.2009

95.1

Mexico

107

0.9948

0.02149

0.73488

14.07.2011

98.7

UK

53.4

0.9864

0.09821

0.4632

02.10.2009

98.2

USA

303.8

1.0075

0.03844

7.078

26.02.2010

99.1

The World

9756

0.9733

0.05028

7.1198

19.02.2010

99.7

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Theoretical Biology and Medical Modelling2009,6:23

resistant (L). In the simplest case, people become resistant after infection. The number of infected people increases proportionally because infected people make contact with sensitive people, and decreases proportionally owing to the number of infected people who become convalescent. The equations for such a model are well-known [4]:

dN

= b’ NM− b’’ N

dt

dM

=−b’ NM

dt

(4)

dL

= b” N

dt

N

+ M+ L= const

where b’ and b” are proportionality coefficients.

For the initial period of epidemic development M>>N, i.e. b’M-b”= b. In this situation, system (4) reduces to equation

(2). Also during the initial period, M<

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Pa n d e m i c H1N1i n f l u e n z a :p r e d i c t i n g t h e c o u r s e o fa pa n d e m i c a n d a s s e s s i n g t h e e f f i c a c y o f t h e p l a n n e d va c c i n at i o n p r o g r a m m e i n t h eU n i t e dS tat e s

S Towers ([email protected].edu)1, Z Feng2

1. Department of Statistics, Purdue University, West Lafayette, Indiana, United States

2. Department of Mathematics, Purdue University, West Lafayette, Indiana, United States

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This article was published on 15 October 2009.

Citation style for this article: Towers S, Feng Z. Pandemic H1N1 influenza: predicting the course of a pandemic and assessing the efficacy of the planned vaccination programme in the United States. Euro Surveill. 2009;14(41):pii=19358. Available online: http://www.eurosurveillance.org/ViewArticle.aspx?ArticleId=19358

We use data on confirmed cases of pandemic influenza A(H1N1), disseminated by the United States Centers for Disease Control and Prevention(US CDC), to fit the parameters of a seasonally forced Susceptible, Infective, Recovered (SIR) model. We use the resulting model to predict the course of the H1N1 influenza pandemic in autumn 2009, and we assess the efficacy of the planned CDC H1N1 vaccination campaign. The model predicts that there will be a significant wave in autumn, with 63% of the population being infected, and that this wave will peak so early that the planned CDC vaccination campaign will likely not have a large effect on the total number of people ultimately infected by the pandemic H1N1 influenza virus.

Introduction

For several years the United States (US) Centers for Disease Control and Prevention (CDC) have had an established protocol for laboratory influenza testing and collection, and dissemination of associated statistics [1]. These statistics are published and regularly updated online [2].

With the recognition of a new, potentially pandemic strain of influenza A(H1N1) in April 2009, the laboratories at the US CDC and the World Health Organization (WHO) dramatically increased their testing activity from week 17 onwards (week ending 2 May 2009), as can be seen in Figure 1. In this analysis, we use the extrapolation of a model fitted to the confirmed influenza A(H1N1) v case counts during summer 2009 to predict the behaviour of the pandemic during autumn 2009.

Methods

The CDC/WHO influenza count data used in these studies were obtained from the weekly online surveillance reports [2]. At the time of writing, the data up to week 38 (week ending 26 September 2009) were the most recent. However, we observed that in each weekly update the data significantly change for at least five weeks prior to the week of the update, likely due to a large backlog in testing. In this analysis we thus used data only up to week 33 (week ending 22 August).

The pandemic potential of influenza A(H1N1)v was recognised during week 16 (week ending 25 April) [3]. We assumed that there was no time bias in the CDC/WHO seasonal influenza count data prior to that date. Based on the extrapolation of the exponential

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decline behaviour of regular seasonal influenza prior to week 16 into the temporal region of heightened testing activity, we found that the data after week 20 (ending 23 May) contain no significant time bias. We thus used the data from week 21 to 33 (from 24 May to 22 August 2009).

The behaviour of the H1N1 influenza pandemic over time was modelled using a seasonally forced deterministic Susceptible, Infective, Recovered (SIR) model [4]:

dS/dt=-β(t) SI/N

(1)

dI/dt=β(t) SI/N – γI,

(2)

where N=305,000,000*.

We assumed that γ=1/3 days-1 [5], and that the contact rate, β(t), was periodically forced via

β(t)=β0+β1 cos(2πt) (3)

The reproduction number was given by R0=β(t)/γ.

To simulate the time evolution of the influenza H1N1 pandemic, we assumed an initial number of infective individuals and susceptibles, I0=1* and S0=N, respectively, at an initial time t0. Given particular values of β0, β1, and t0, we numerically solved equations (1) and (2) to estimate the fraction of the population infected with pandemic H1N1 influenza each week.

We compared the shape of the results of the deterministic model to the shape of the actual pandemic influenza data, and found the parameters {β0,β1,t0} that provided the best Pearson chi-square statistics.

The grid search for the parameters that minimised the chi-square value was performed with parameter ranges:

β0 between 0.92γ to 2.52γ in increments of 0.02γ,*

β1 between 0.05γ to 0.80γ in increments of 0.01γ, and* t0 between weeks -8 to 10 (relative to the beginning of 2009),

in increments of one week.

The planned CDC vaccination programme against pandemic H1N1 influenza will begin with six to seven million doses being delivered by the end of the first full week in October (week 40), with 10 to 20 million doses being delivered weekly thereafter [6]. We included the effects of this vaccination campaign into

www. eurosurveillance . org 1

our seasonally forced SIR model by decreasing the number of susceptibles in the population by the corresponding amounts. For healthy adults, full immunity to H1N1 influenza is achieved about two weeks after vaccination with one dose of the vaccine [7,8], and

F i g u r e 1

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Influenza-positive tests reported to the US CDC by US WHO/ NREVSS-collaborating laboratories, national summary, United States, 2009 until 26 September

6,000

specimens

5,000

4,000

of positive

3,000

Number

2,000

1,000

0

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

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