# Modelling and analysis of influenza A (H1N1) on networks

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## Abstract

### Background

In April 2009, a new strain of H1N1 influenza virus, referred to as pandemic influenza A (H1N1) was first detected in humans in the United States, followed by an outbreak in the state of Veracruz, Mexico. Soon afterwards, this new virus kept spreading worldwide resulting in a global outbreak. In China, the second Circular of the Ministry of Health pointed out that as of December 31, 2009, the country’s 31 provinces had reported 120,000 confirmed cases of H1N1.

### Methods

We formulate an epidemic model of influenza A based on networks. We calculate the basic reproduction number and study the effects of various immunization schemes. The final size relation is derived for the network epidemic model. The model parameters are estimated via least-squares fitting of the model solution to the observed data in China.

### Results

For the network model, we prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction is less than one. The final size will depend on the vaccination starting time, *T*, the number of infective cases at time *T* and immunization schemes to follow. Our theoretical results are confirmed by numerical simulations. Using the parameter estimates based on the observation data of the cumulative number of hospital notifications, we estimate the basic reproduction number *R*_{0} to be 1.6809 in China.

### Conclusions

Network modelling supplies a useful tool for studying the transmission of H1N1 in China, capturing the main features of the spread of H1N1. While a uniform, mass-immunization strategy helps control the prevalence, a targeted immunization strategy focusing on specific groups with given connectivity may better control the endemic.

## Keywords

Influenza Degree Distribution Spectral Radius Reproduction Number Epidemic Model## List of abbreviations used

- H1N1
Swine Influenza A

- WHO
World Health Organization

- CDC
Center for Disease Control

- GAS
Global Asymptotic Stability

- SF
Scale-Free.

## Introduction

The H1N1 pandemic calls for action, and various mathematical models have been constructed to study the spread and control of H1N1. Fraser *et al.* estimated the basic reproduction number *R*_{0}[5] in the range of 1.4 to 1.6 by analyzing the outbreak in Mexico, and earlier data of the global spread [6]. Nishiura *et al.* also estimated the reproduction number *R*_{0} but in the range of 2.0 to 2.6 for Japan [7]; they also estimated the reproduction number as 1.96 for New Zealand [8]. Vittoria Colizza *et al.* used a global epidemic and mobility model to obtain the estimation of the size of the epidemic in Mexico as well as that of imported cases at the end of April, 2009 [9]. Marc Baguelin *et al.* presents a real-time assessment of the effectiveness and cost-effectiveness of alternative influenza A (H1N1) vaccination strategies by a dynamic model [10]. H1N1, like many other infectious diseases, is intrinsically related to human social networks; it exhibits great heterogeneity in terms of the numbers and the pattern of contacts. The usual compartmental modelling in epidemiology generally assumes that population groups are fully and homogeneously mixed, but this does not reflect the real situation of the variation in the process of contact transmission. The epidemic modelling on complex networks has been attracting great interest, and various epidemic models on complex networks have been extensively investigated in recent years [11, 12, 13, 14, 15, 16, 17].

## The network model and parameters

*S*), exposed (

*E*), asymptomatically infected (

*A*), symptomatically infected (

*I*) and removed/immune (

*R*)

*.*The asymptomatically infected compartment contains those who fail to show noticeable symptoms or with light flu-like symptoms; they are not identified as H1N1 cases, but are able to spread the infection. We assume that a susceptible individual becomes infected if they come into contact with an asymptomatically or symptomatically infective individual. Then, the susceptible enters the exposed class

*E*of those in the latent period. The period of incubation for H1N1 is 1-3 days [3]. After the latent period, the individual enters the class

*I*or

*A*of infectives, who are infectious in the sense that they are capable of transmitting the infection. When the infectious period ends, the individual enters the recovered class

*R.*We assume that a removed individual will never become susceptible or infected again. In our model, new births, natural deaths and migrations are ignored. The flow diagram of the individuals is depicted in Figure 2.

*n*distinct groups of sizes

*N*

_{ k }(

*k*= 1, 2, …,

*n*) such that each individual in group

*k*has exactly

*k*contacts per day. If the whole population size is

*N*(

*N*=

*N*

_{1}

*+ N*

_{2}+ ⋯ +

*N*

_{ n }), then the probability that a uniformly chosen individual has

*k*contacts is

*P*(

*k*) =

*N*

_{ k }/

*N*, which is called the degree distributions of the network. Empirical studies have shown that many real networks have scale-free (SF) degree distributions

*P*(

*k*)

*≈ k*

^{ –γ }with 2

*≤*γ

*≤*3 where the epidemic model does not show an epidemic threshold (see [18]) and Poisson degree distributions

*P*(

*k*) =

*µ*

^{ k }/

*k*! exp(

*–µ*) (see [19]). If

*S*

_{ k },

*E*

_{ k },

*A*

_{ k },

*I*

_{ k }and

*R*

_{ k }represent the number of susceptible, exposed, asymptomatically infected, symptomatically infected and recovered individuals within group

*k*(where

*S*

_{ k }+

*E*

_{ k }+

*A*

_{ k }+

*I*

_{ k }+

*R*

_{ k }=

*N*

_{ k }), then the following system of differential equations captures disease spread for arbitrarily large networks (

*N → ∞*), for both transmission through the network and the mean-field type transmission

where Open image in new window represent the expectation that any given edge points to an infected and asymptomatically infected vertex respectively. Note that Open image in new window ; thus, *S*_{ k }(*t*) + *E*_{ k }(*t*) + *A*_{ k }(*t*) + *I*_{ k }(*t*) + *R*_{ k }(*t*) = *N*_{ k } is constant.

The densities of susceptible, exposed, asymptomatically infected, symptomatically infected and recovered nodes of degree *k* at time *t*, are denoted by *s*_{ k }, *e*_{ k }, *a*_{ k }, *i*_{ k } and *r*_{ k }, respectively. If *S*_{ k }, *E*_{ k }, *A*_{ k }, *I*_{ k } and *R*_{ k } are used to represent *s*_{ k }, *e*_{ k }, *a*_{ k }, *i*_{ k }, and *r*_{ k } respectively, we can still use system (1)-(5) to describe the spread of disease on the network. Clearly, these variables obey the normalization condition

Parameters of the model

Parameters | description |
---|---|

| transmission coefficient between community |

| transmission coefficient between community |

| rate of becoming infectious after latentcy |

| rate of becoming asymptomatically infected |

1 | rate of becoming symptomatically infected |

| recovery rate of asymptomatically infected |

| recovery rate of symptomatically infected |

The mathematical formulation of the epidemic modelling on the network is completed with the initial conditions given as *S*_{ k }(0) = *S*_{ k }_{0}, *I*_{ k }(0) = *I*_{ k }_{0}, *E*_{ k }(0) = *A*_{ k }(0) = *R*_{ k }(0) = 0*.*

## Analysis

### Stability and basic reproduction number

*P*

^{0}(1, ⋯ , 1, ⋯ ,1, 0, 0, ⋯ , 0). Following van den Driessche and Watmough [20], we note that only compartments

*E*

_{ k },

*A*

_{ k }and

*I*

_{ k }are involved in the calculation of

*R*

_{0}. In the infection-free state

*P*

^{0}, the rate of appearance of new infections

*F*and the rate of transfer of individuals out of the two compartments

*V*are given by

Using the concepts of next-generation matrix [20], the reproduction number is given by *R*_{0} = *ρ*(*FV*^{ – }^{1}), the spectral radius of the matrix *FV*^{ – }^{1}.

*FV*

^{ – }

^{1}, we first represent the inverse of

*V*by the following matrix:

Now we are ready to compute the eigenvalues of the matrix *C* = *FV*^{ – }^{1}.

*C*and Open image in new window have the same spectral radius. Since matrix Open image in new window has rank 1, the spectral radius Open image in new window is equal to the trace of Open image in new window . Note that

In summary, we have the following theorem.

**Theorem 1** If *R*_{0} < 1, the infection-free equilibrium *P*^{0}(1, ⋯ , 1, ⋯ , 1, 0, 0, ⋯ , 0) of system (1)-(5) is locally asymptotically stable, and if *R*_{0} > 1 the infection-free equilibrium *P*^{0} is unstable.

Next, we will prove the global asymptotic stability of the infection-free equilibrium.

**Theorem 2** If *R*_{0} < 1, the infection-free equilibrium *P*^{0}(1, ⋯ , 1, ⋯ , 1, 0, 0, ⋯ , 0) of system (1)-(5) is global asymptotically stable.

**Proof.**Let us consider the Lyapunov function of the form:

where Open image in new window .

*L*(

*t*) along the solutions of system (1)-(5). It is seen that

Furthermore, *L′*(*t*) = 0 only if *A*_{ k } = *I*_{ k } = 0. Therefore, the global stability of *P*^{0} when *R*_{ 0 } < 1 follows from LaSalle’s Invariance Principle [21].

## Estimation of parameters

*i*(

*t*) [22, 23, 24]. In other words, we are looking for the set of parameters Λ = (λ

_{1}, λ

_{2}, γ, δ, α

_{1}, α

_{2}) such that the associated model solution best fits the epidemic data by minimizing the sum of the squared differences between the observed data

*i*(

*t*) and the total number Open image in new window . Therefore, we need to minimize the objective function:

where *n*_{ d } represents the number of days we choose from the observed data.

In the real world, *P*(*k*) usually obeys a power-law distribution. Hence, *P*(*k*) = 2*m*^{ 2 }*k*^{ –ν } (*m* = 3 and *ν* = 3*.*5) is used in model (1)-(5).

*t*= 0

*.*04. By using Euler and the advanced alternate directions scheme [22], we estimate the parameters and summarize them in Table 2. Using the parameters in Table 1, a straightforward computation using formula (6) gives the basic reproductive number for the H1N1 epidemic in China as

*R*

_{0}= 1

*.*6809. Using the parameters in Table 2, we compared the model simulation results with the observation data in Figure 3. It can be seen that our model captures the main features of the spread of the H1N1 in China.

Parameters estimated from the observed data in China

Parameters | Estimated value |
---|---|

| 0.01 |

| 0.188 |

| 0.4 |

| 0.85 |

| 0.141 |

| 0.141 |

## The effect of vaccination strategies

Vaccination is very powerful in controlling influenza. In this section, we will discuss the impact of various immunization schemes.

### Uniform immunization strategy

*p*for the immunization rate (0 <

*p*< 1), by substituting

*λ*

_{1}

*→*(1 –

*p*)

*λ*

_{1}and

*λ*

_{2}

*→*(1 –

*p*)

*λ*

_{2}in model (1)-(5), the model becomes

We obtain the critical fraction *p*_{ c } for the prevention and control of the prevalence of H1N1 as Open image in new window . For the case of China, this is Open image in new window . In other words, in order to control the prevalence, at least 40% of the whole susceptible population would have to be immunized through vaccination (about 536 million individuals).

### Targeted immunization

*κ*

_{1}and

*κ*

_{2}, such that if

*k*>

*κ*

_{2}, all nodes with connectivity

*k*are immunized, while if

*κ*

_{1}<

*k*<

*κ*

_{2},

*p*

_{ k }(0 <

*p*

_{ k }

*≤*1) portion will be immunized, and

*p*

_{ k }is defined as the fraction of individuals to be immunized, i.e., we define the immunization rate

*σ*

_{ k }as

*p*

_{ k }=

*p*in model (7). We plot

*R*

_{0}as a function of

*k*

_{2}and

*p*in Figure 4. One can see from this figure that

*R*

_{0}is an increasing function of

*k*

_{2}but a decreasing function of

*p.*In other words, if

*p*is large or

*k*

_{2}is small, more people receive vaccination, then H1N1 can be controlled.

## The final size relation

First, we show that for the model (1)-(5) the disease will eventually die out, i.e., *A*(∞) = 0, *E*(∞) = 0, and *I*(∞) = 0.

*S*

_{ k },

*A*

_{ k },

*I*

_{ k },

*R*

_{ k }≥ 0 and

*S*

_{ k }

*+ A*

_{ k }

*+ I*

_{ k }

*+ R*

_{ k }=

*1.*Observing that

*S*

_{ k }(

*t*) +

*E*

_{ k }(

*t*) is decreasing whenever

*E*

_{ k }> 0. However,

*S*

_{ k }+

*E*

_{ k }is bounded below by 0; hence, it has a limit. Moreover, model (1)-(5) implies that Open image in new window is bounded because

*E*

_{ k }(

*t*) is bounded. Hence Open image in new window , so

*E*

_{ k }(

*∞*) = 0. Similarly, we can prove that

*A*(

*∞*) = 0 and

*I*(

*∞*) = 0. We adopt the convention that, for an arbitrary continuous function

*w*(

*t*) with non-negative components, Open image in new window . If we integrate the seventh equation from

*t*= 0 to

*∞*, we have

*S*

_{ k }(0),

*S*

_{ k }(

*∞*),

*E*

_{ k }(0) and

*E*

_{ k }(

*∞*) are bounded by the initial total population size. Therefore, the right-hand side of (10) is also finite and

*δ*is positive. Since

*E*

_{ k }(

*∞*) = 0, we have

### The final size without vaccination

*t → ∞*, we have

*S*

_{ k }(0) =

*S*

_{ k }

_{0},

*I*

_{ k }(0) =

*I*

_{ k }

_{0},

*E*

_{ k }(0) =

*A*

_{ k }(0) = 0, then the final size relation becomes

*S*

_{ k }(0) =

*S*

_{ k }

_{0},

*I*

_{ k }(0) =

*I*

_{ k }

_{0},

*E*

_{ k }(0) =

*E*

_{ k }

_{0},

*A*

_{ k }(0) =

*A*

_{ k }

_{0}, then the final size relation becomes

### The final size with vaccination

To fully see the effect of vaccination, we show that the final size of susceptible, recovered and vaccinated individuals. It can be seen from Figure 5 that the final size of the susceptible and vaccinated increase as *p* increases. However, the final size of the recovered is a decreasing function of *p*.

### The final size with vaccination from time *T*

*T*follows a targeted immunization scheme, integration of equation (1) from 0 to

*T*and integration of equation (8) from

*T*to

*t*(

*t*>

*T*) gives

*t → ∞*, we can obtain the final size relation with targeted immunization scheme from time

*T*

## Conclusions

Network models can capture the main features of the spread of the H1N1. In this paper, using a network epidemic model for influenza A (H1N1) in China, we calculated the basic reproduction number *R*_{0} and discussed the local and global dynamical behaviors of the disease-free equilibrium. The effects of various immunization schemes were studied and compared. A final size relation was derived for the network epidemic models. The derivation depends on an explicit formula for the basic reproduction number of network disease transmission models. The transmission coefficients are estimated through least-squares fitting of the model to observed data of the cumulative number of hospital notifications. We also gave the estimated value for the reproduction number for influenza A (H1N1) in China as *R*_{0} = 1*.*6809.

Parameters were estimated during the period when the vaccination was not applied. For these parameters, we found that *γ* = 0*.*85, which means that 15% of the exposed become infected during the early course of the endemic. Although vaccination commenced in China in November 2009, we were not able to compare the real data with the model projections due to lack of data.

## Notes

### Acknowledgements

This article has been published as part of *BMC Public Health* Volume 11 Supplement 1, 2011: Mathematical Modelling of Influenza. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2458/11?issue=S1.

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