Modelling and analysis of influenza A (H1N1) on networks
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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.
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.
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 R0 to be 1.6809 in China.
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.
KeywordsInfluenza Degree Distribution Spectral Radius Reproduction Number Epidemic Model
List of abbreviations used
Swine Influenza A
World Health Organization
Center for Disease Control
Global Asymptotic Stability
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 R0 in the range of 1.4 to 1.6 by analyzing the outbreak in Mexico, and earlier data of the global spread . Nishiura et al. also estimated the reproduction number R0 but in the range of 2.0 to 2.6 for Japan ; they also estimated the reproduction number as 1.96 for New Zealand . 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 . 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 . 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
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
transmission coefficient between community S k and A i
transmission coefficient between community S k and I i
rate of becoming infectious after latentcy
rate of becoming asymptomatically infected
1 – γ
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.
Stability and basic reproduction number
Using the concepts of next-generation matrix , the reproduction number is given by R0 = 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 R0 < 1, the infection-free equilibrium P0(1, ⋯ , 1, ⋯ , 1, 0, 0, ⋯ , 0) of system (1)-(5) is locally asymptotically stable, and if R0 > 1 the infection-free equilibrium P0 is unstable.
Next, we will prove the global asymptotic stability of the infection-free equilibrium.
Theorem 2 If R0 < 1, the infection-free equilibrium P0(1, ⋯ , 1, ⋯ , 1, 0, 0, ⋯ , 0) of system (1)-(5) is global asymptotically stable.
where Open image in new window .
Furthermore, L′(t) = 0 only if A k = I k = 0. Therefore, the global stability of P0 when R 0 < 1 follows from LaSalle’s Invariance Principle .
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 power-law 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
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 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).
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
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 R0 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 R0 = 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.
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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