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This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The gravity model is often used in predicting the spread of influenza. We use the data of influenza A (H1N1) to check the model’s performance and validation, in order to determine the scope of its application. In this article, we proposed to model the pattern of global spread of the virus via a few important socio-economic indicators. We applied the epidemic gravity model for modelling the virus spread globally through the estimation of parameters of a generalized linear model. We compiled the daily confirmed cases of influenza A (H1N1) in each country as reported to the WHO and each state in the USA, and established the model to describe the relationship between the confirmed cases and socio-economic factors such as population size,

Influenza A (H1N1) is one of the most common virus strains causing influenza pandemics in humans [

During the spread of influenza, spatial waves of infection have been observed between large distant populations [

The gravity model considers the effect of distance and the size of the donor and recipient communities [_{ij}_{i}_{j}_{1}_{2}_{ij}_{i}_{i}_{i}_{i}_{1}_{2}_{3}

We used a generalized linear model (GLM) [_{i}_{0}_{1}_{2}_{3}_{1}_{2}_{3}

We compared the performance of the gravity model at two spatial scales: global spread and national spread in the USA, assuming a single source of the virus,

We downloaded

The GLM demonstrated that, in log-scale, the number of daily cumulative confirmed cases of influenza A (H1N1) was statistically significantly associated (positively) with population size, except for 28 April and

Using the regressed coefficients of GLM for the day of 6 July 2009, we obtained the gravity model to estimate cases N of influenza A (H1N1) in each country i (omitting the error terms):

The value and standard errors of the model parameters for variables ln(intercept), ln(G), ln(P), and ln(D) are 3.44 ± 1.496, 1.547 ± 0.111, 1.575 ± 0.113, and 2.108 ± 0.233, respectively. Our estimation of the number of confirmed influenza A (H1N1) cases in each country (

For each country, we compared the number of predicted cases from the model and reported confirmed cases based on the data on 6 July 2009 (

When we used the number of days since 23 April 2009 to the first confirmed infection for each country as the dependent variable in

We compared the number of predicted days and observed days (_{i}) and larger population size (P_{i}) would lead to a shorter waiting time to the first confirmed case and longer distance (D_{i}) would lead to a longer waiting time.

Our results showed that the spread of influenza A (H1N1) among countries was significantly associated to covariates of a set of important socio-economic indicators. The results were consistent with previous findings that air and surface transportation played a significant role in the spread of influenza under both epidemiological survey (e.g., [

We modified the epidemic gravity model with the assumption of a surrogate origin (

The significance of each covariate (

The daily cumulative confirmed cases of influenza A (H1N1) was used in our analysis, but these cases may not represent the true prevalence of the infection in each region. The number of cases identified was clearly related to the effort and the resources devoted by the health agencies in a country. For a new infectious disease, it is very likely that many cases probably existed already in many parts of the world before the identification of the first case. This is especially true due to the modern transportation systems and possibly many symptomatic and asymptomatic carriers have travelled to many places outside the borders already before the identification of the cases. Following the extensive media reports right after the first identification of the new subtype of the virus, many countries had increased the screening on border-crossing population without paying much attention to their domestic populations at the beginning of the new influenza A (H1N1) 2009 surveillance. The effort of screening only symptomatic cases or their close contacts of confirmed cases entering the country would result finding the cases from a small and biased sample [

The three covariates in the model were selected the availability and their important roles in global social and economic interactions. GDP represents the economic activity of the people (for international travel), population size represents the susceptible, and distance represents a possible barrier to infection. Our GLM model provides a quantitative method to estimating the parameters in the model. The model we used was heuristic through conceptual reasoning, but the method of finding the parameters in the model was based on statistical estimation. Mathematical and statistical modelling is an important aspect in addressing public health challenges [

This work is supported by Chinese Academy of Sciences (CAS) (Knowledge Innovation Project, Grant No. KSCX2-YW-R-162 to X.L. and KSCX2-YW-N-063 to F.L.), Chinese Ministry of Science and Technology (Technical Platform Program), and United States Department of Agriculture (Cooperative Project of Surveillance on Avian Influenza).

The p-values for testing the significance of the covariates (log-transformed population size (

The observed (

(