Volume 11 Supplement 1
Mathematical Modelling of Influenza
Modelling and analysis of influenza A (H1N1) on networks
 Zhen Jin^{1},
 Juping Zhang^{1},
 LiPeng Song^{1},
 GuiQuan Sun^{1},
 Jianli Kan^{2} and
 Huaiping Zhu^{3}Email author
DOI: 10.1186/1471245811S1S9
© Jin et al; licensee BioMed Central Ltd. 2011
Published: 25 February 2011
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 leastsquares fitting of the model solution to the observed data in China.
Results
For the network model, we prove that the diseasefree 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, massimmunization strategy helps control the prevalence, a targeted immunization strategy focusing on specific groups with given connectivity may better control the endemic.
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 realtime assessment of the effectiveness and costeffectiveness 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 [11–17].
The network model and parameters
where represent the expectation that any given edge points to an infected and asymptomatically infected vertex respectively. Note that ; 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 

λ _{1}  transmission coefficient between community S_{ k } and A_{ i } 
λ _{2}  transmission coefficient between community S_{ k } and I_{ i } 
δ  rate of becoming infectious after latentcy 
γ  rate of becoming asymptomatically infected 
1 – γ  rate of becoming symptomatically infected 
α _{1}  recovery rate of asymptomatically infected 
α _{2}  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
Using the concepts of nextgeneration matrix [20], the reproduction number is given by R_{0} = ρ(FV^{ – }^{1}), the spectral radius of the matrix FV^{ – }^{1}.
Now we are ready to compute the eigenvalues of the matrix C = FV^{ – }^{1}.
In summary, we have the following theorem.
Theorem 1 If R_{0} < 1, the infectionfree equilibrium P^{0}(1, ⋯ , 1, ⋯ , 1, 0, 0, ⋯ , 0) of system (1)(5) is locally asymptotically stable, and if R_{0} > 1 the infectionfree equilibrium P^{0} is unstable.
Next, we will prove the global asymptotic stability of the infectionfree equilibrium.
Theorem 2 If R_{0} < 1, the infectionfree equilibrium P^{0}(1, ⋯ , 1, ⋯ , 1, 0, 0, ⋯ , 0) of system (1)(5) is global asymptotically stable.
where .
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
where n_{ d } represents the number of days we choose from the observed data.
In the real world, P(k) usually obeys a powerlaw distribution. Hence, P(k) = 2m^{ 2 }k^{ –ν } (m = 3 and ν = 3.5) is used in model (1)(5).
Parameters estimated from the observed data in China
Parameters  Estimated value 

λ _{ 1 }  0.01 
λ _{ 2 }  0.188 
δ  0.4 
γ  0.85 
α _{ 1 }  0.141 
α _{ 2 }  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
We obtain the critical fraction p_{ c } for the prevention and control of the prevalence of H1N1 as . For the case of China, this is . 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
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.
The final size without vaccination
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
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 diseasefree 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 leastsquares 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.
List of abbreviations used
 H1N1:

Swine Influenza A
 WHO:

World Health Organization
 CDC:

Center for Disease Control
 GAS:

Global Asymptotic Stability
 SF:

ScaleFree.
Declarations
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/14712458/11?issue=S1.
Authors’ Affiliations
References
 WHO: Pandemic (H1N1) 2009  update 82. [http://www.who.int/csr/don/20100108/en/]
 H1N1 Public Files. [http://www.moh.gov.cn/publicfiles/business/htmlfiles/mohwsyjbgs/s7863/201001/45434.htm]
 Updated Interim Recommendations for the Use of Antiviral Medications in the Treatment and Prevention of Influenza for the 20092010 Season. [http://www.cdc.gov/h1n1flu/recommendations.htm]
 Bronze MS: H1N1 Influenza (Swine Flu). [http://emedicine.medscape.com/article/1807048overview]
 Heffernan JM, Smith RJ, Wahl LM: Perspectives on the basic reproductive ratio. J. R. Soc. Interface. 2005, 2 (4): 281293. 10.1098/rsif.2005.0042.PubMed CentralView ArticlePubMed
 Fraser C, Donnelly CA, Cauchemez S, William P Hanage, M DVK, Hollingsworth TD, Griffin J, Baggaley RF, Jenkins HE, Lyons EJ, Jombart T, Hinsley WR, Grassly NC, Balloux F, Ghani AC, Ferguson NM, Rambaut A, Pybus OG, LopezGatell H, AlpucheAranda CM, Chapela IB, Zavala EP, Guevara DME, Checchi F, Garcia E, Hugonnet S, Roth C: Pandemic Potential of a Strain of Influenza A(H1N1): Early Findings. Science. 2009, 324: 15571561. 10.1126/science.1176062.PubMed CentralView ArticlePubMed
 Nishiura H, CastilloChavez C, Safan M, Chowell G: Transmission potential of the new influenza A(H1N1) virus and its agespecificity in Japan. Eurosurveillance. 2009, 14: p11=19227
 Hiroshi N, Wilson N, Baker MG: Estimating the reproduction number of the novel influenza A virus (H1N1) in a Southern Hemisphere setting: preliminary estimate in New Zealand. J New Zealand Medical Association. 2009, 122: 7377.
 Colizza V, Vespignani A, Perra N, Poletto C, Gonalves B, Hu H, Balcan D, Paolotti D, den Broeck WV, Tizzoni M, Bajardi P, Ramasco JJ: Estimate of Novel Influenza A/H1N1 cases in Mexico at the early stage of the pandemic with a spatially structured epidemic model. PLoS Curr Influenza. 2009, 11: RRN112910.1371/currents.RRN1129.
 Baguelin M, Hoek AJV, Jit M, Flasche S, White PJ, Edmunds WJ: Vaccination against pandemic influenza A/H1N1v in England: A realtime economic evaluation. Vaccine.
 PastorSatorras R, Vespignani A: Epidemic Spreading in ScaleFree Networks. Phys Rev Lett. 2001, 86: 32003203. 10.1103/PhysRevLett.86.3200.View ArticlePubMed
 PastorSatorras R, Vespignani A: Epidemic dynamics and endemic states in complex networks. Phys Rev Lett. 2001, 63: 066117
 Moreno Y, PastorSatorras R, Vespignani A: Epidemic outbreaks in complex heterogeneous networks. Eur Phys J B. 2002, 26: 521529.
 Madar N, Kalisky T, Cohen R, ben Avraham D, Havlin S: Immunization and epidemic dynamics in complex networks. Eur Phys J B. 2004, 38: 269276. 10.1140/epjb/e2004001198.View Article
 Yang R, Wang BH, Ren J, Bai WJ, Shi ZW, Wang WX, Zhou T: Epidemic spreading on heterogeneous networks with identical infectivity. Phys Lett A. 2007, 364: 189193. 10.1016/j.physleta.2006.12.021.View Article
 Volz E: SIR dynamics in random networks with heterogeneous connectivity. J Math Biol. 2008, 56: 293310. 10.1007/s0028500701164.View ArticlePubMed
 GomezGardenes J, Latora V, Moreno Y, Profumo E: Spreading of sexually transmitted diseases in heterosexual populations. PNAS. 2008, 105: 13991404. 10.1073/pnas.0707332105.PubMed CentralView ArticlePubMed
 Albert R, Barabási A: Statistical mechanics of complex networks. Rev Mod Phys. 2002, 74: 4797. 10.1103/RevModPhys.74.47.View Article
 Meyers LA, Pourbohloul B, Newman M, Skowronski DM, Brunham RC: Network theory and SARS: predicting outbreak diversity. J Theor Biol. 2005, 232: 7181. 10.1016/j.jtbi.2004.07.026.View ArticlePubMed
 van den Driessche P, Watmough J: Reproduction numbers and sub threshold endemic equilibria for compartmental models of disease transmission. Math Bios. 2002, 180: 2948. 10.1016/S00255564(02)001086.View Article
 Hale JK: Ordinary differential equations. New York: WileyInterscience. 1969
 Chowell G, Nishiura H: Quantifying the transmission potential of pandemic influenza. Phys Life Rev. 2008, 5: 5077. 10.1016/j.plrev.2007.12.001.View Article
 Chowell G, Ammon C, Hengartner N, Hyman J: Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions. J Theor Biol. 2006, 241: 193204. 10.1016/j.jtbi.2005.11.026.View ArticlePubMed
 Sertsou G, Wilson N, Baker M, Nelson P, Roberts MG: Key transmission parameters of an institutional outbreak during the 1918 influenza pandemic estimated by mathematical modelling. Theor Biol Med Model. 2006, 3: 3844. 10.1186/17424682338.PubMed CentralView ArticlePubMed
 PastorSatorras R, Vespignani A: Immunization of complex networks. Phys Rev E. 2002, 65: 03610410.1103/PhysRevE.65.036104.View Article
 Fu X, Small M, Walker DM, Zhang H: Epidemic dynamics on scalefree networks with piecewise linear infectivity and immunization. Phys Rev E. 2008, 77: 03611310.1103/PhysRevE.77.036113.View Article
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