This work is for an academic and research purpose and should not constitute a decision making support. We used a model that is simple in its structure and its assumptions. The results do not necessarily reflect the reality. Its purpose is to illustrate how social structure and age distribution might affect the dynamic and public health consequences of an outbreak. The codes and analyses used for this work are available on github.

Introduction

“assessing the effectiveness of interventions that specifically target social networks, such as school closure [or lock-down], requires models that explicitly account for such social structure” (Prem et al. 2017).
The objective of the present work was to investigate, in absence of any intervention, how much country-specific age distribution and social contact pattern affect the basic reproduction number (\(R_{0}\)) and the expected spread of the COVID-19 disease. Using an age-structured SIR model fed with age distribution data and contact matrices, we predicted and mapped \(R_{0}\) for 147 countries. We also present the dynamic of the outbreak and age distribution of hospitalization, as predicted by the model, for four selected countries.


Methods

The SIR model

We used an age-structured SIR model with 16 age classes (0-4; 5-9; 10-14; … 75+). It allow to explicitly account for the social structure of the studied population. \[\begin{align} \frac{dS_{i}}{dt} = -\beta S_{i} \sum_{j} C_{ij} I_{j}/N_{i} \end{align}\] \[\begin{align} \frac{dI_{i}}{dt} = \beta S_{i} \sum_{j} C_{ij} I_{j}/N_{i} - \gamma I_{i} \end{align}\] \[\begin{align} \frac{dR_{i}}{dt} = \gamma I_{i} \end{align}\]
Where \(\beta\) is the probability of transmission on contact, \(i\) and \(j\) are indices for age classes, \(\gamma\) is the recovery rate. \(C\) is the contact matrix. \(N_{j}\) is the population in age \(j\).


Data

Contact matrices for 152 countries were obtained from the work of Prem et al. (2017).
Data of age structure were obtained from the Health Nutrition and Population (HNP) database available on the World Bank website. Age stratified estimates of infection severity were taken from the article of Verity et al. (2020).

Model parameters

From an estimated \(R_{0}\) of 2.5 for France (ETE modelling team, 2020), we estimated parameter \(\beta\) (probability of transmission on contact) using the following equality (Towers et al. 2012): \[\begin{align} \beta = \frac{R_{0}}{\gamma}\lambda \end{align}\] Where \(\lambda\) is the largest eigenvalue of the matrix \(M\) such as \(M_{ij} = C_{ij}f_{i}/f{j}\) with \(f_{i}\) and \(f_{j}\) are frequencies in the population of age classes \(i\) and \(j\), respectively. Parameter \(\gamma\) (recovery rate) was set to \(\frac{1}{14}\) days\(^{-1}\).

Analyses

All analyses were performed using the software R. The script for the model was adapted from here to be used with more than 2 age classes.
We calculated \(R_{0}\) for countries having both contact matrix and age structure data available using the value of \(\beta\) calculated with France data (see Model Parameter paragraph). Predicted \(R_{0}\) for 147 countries were mapped using additional R packages sf and plotly.
We simulated the dynamic of the outbreak (prevalence of infection and prevalence of immunization in the population) for 4 selected countries: France, Germany, China and Niger. For this task, we introduced in the model an infected individual in the 40-44 age group.
We used age stratified proportion of severe infections to predict the age distribution of severe cases.
Data and codes are available online at GitHub.

Results

Predicting and mapping \(R_{0}\) worldwide

Probability \(\beta\) of transmission on contact based on an estimated \(R_{0}\) of 2.5 in France was estimated to be 0.0254205. Using this value of \(\beta\), we were able to predict \(R_{0}\) in 147 countries (Figure 1) for which both contact matrices and age structure data were available.

Figure 1: Map of predicted \(R_{0}\) for 147 countries in absence of intervention according to an age-structured SIR model.

Countries differ only for age distribution of their population and for social contact patterns. Other parameters (\(\beta\) and \(\gamma\)) being equals.These \(R_{0}\) values are prediction of a model, they are not real data

Predicted \(R_{0}\) ranged from 1.52 in Germany to 5.65 in Niger. Higher values of \(R_{0}\) were found to be in the inter-tropical region.

Age distribution and contact characteristics of four selected countries

We selected four countries: Niger (highest predicted \(R_{0}\)), France, China and Germany (lowest predicted \(R_{0}\)). Age distribution and contact matrix for each of these countries are presented in Figure 2.



Figure 2: Age distribution, daily number of contacts and contact matrices for Niger (A, E, I), France (B, F, J), China (C, G, K) and Germany (D, H, L).

Horizontal red dashed lines indicate the mean number of daily contacts per individuals. For contact matrix plots, darker color intensities indicate more contacts. Data from here and here.


Age distribution of Niger is typical of countries with high birth rate and low life expectancy, the size of each cohort is larger than that of older cohorts. In average, individuals in Niger experience 24 c.d\(^{-1}\) (contacts per days). These contacts occur mostly between individuals of < 20 years old (that represent more than 60% of the total population).
Age distribution of France is almost stationary with a somewhat equal proportion of the population in each age group. The mean number of daily contact is 11.8 c.d\(^{-1}\) mostly with individual of the same age. There is little differences of daily number of contacts among age classes.
Age distributions of China and Germany are constrictive with lower percentage of younger people. The mean number of daily contact in China was 14 c.d\(^{-1}\), closed to what is observed in France but with more differences between age classes. Individuals in Germany experience the lowest number of daily contact with a mean of 6.9 c.d\(^{-1}\).


Outbreak simulation for Niger, France, China and Germany

We then simulated the outbreak in these four countries (Niger, France, China and Germany) using the SIR model. Predicted \(R_{0}\) for these countries were as follow:

## Niger R0 = 5.649338 
## France R0 = 2.499001 
## China R0 = 2.978447 
## Germany R0 = 1.514464

Reminder: \(R_{0}\) of France was not predicted but estimated here and we used this value to estimate probability \(\beta\) of transmission on contact that was used, in turn, to estimate \(R_{0}\) of other countries (see Methods section).

Here, we see that the higher is the mean number of contact per individuals, the higher is \(R_{0}\). Higher \(R_{0}\) is expected to induce faster outbreaks and a higher proportion of infected (and then immunized) individuals (Figure 3 and 4).

Figure 3: Prevalence of infection predicted by an age-structured SIR model fed with population data of Niger, France, China and Germany for an uncontrolled Covid19 epidemic.


Figure 4: Prevalence of immunization predicted by an age-structured SIR model fed with population data of Niger, France, China and Germany for an uncontrolled Covid19 epidemic.



Total severe cases and their distribution by ages

We used estimates of infection severity (i.e. needing hospitalization) provided by Verity et al. (2020) to predict overall proportions of the population expected to need hospitalization due to Covid-19 and its age distribution (Figure 5).

The proportions of the population that were expected to need an hospitalization were 1.3 %, 4.2 %, 3.8 % and 2.3 % in Niger, France, China and Germany, respectively.


Figure 5: Expected age distribution of severe cases during an uncontrolled Covid19 outbreak for Niger (A), France (B), China (C) and Germany (D).



Despite a higher proportion of the population expected to be infected, Niger was predicted to experience the lower number (in proportion) of severe case among the four studied countries. However, the median age for severe cases is expected to fall in the 40-50 age group whereas it is expected to be in the 50-60 age group for Germany and China. In France, the median age for severe cases is expected to fall in the 60-70 age group.

References

Prem K, Cook AR, Jit M. Projecting social contact matrices in 152 countries using contact surveys and demographic data. PLOS Computational Biology. 12 sep 2017;13(9):e1005697.

Towers S, Feng Z. Social contact patterns and control strategies for influenza in the elderly. Math Biosci. dec 2012;240(2):241‑9.

Verity R, Okell LC, Dorigatti I, Winskill P, Whittaker C, Imai N, et al. Estimates of the severity of coronavirus disease 2019: a model-based analysis. Lancet Infect Dis. 30 mar 2020.

ETE modelling team. Estimating the basic reproduction number of the COVID-19 epidemic in France. http://alizon.ouvaton.org/Report1_R0_France.html. 14 mar 2020.


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