Insights into the evolution and emergence of a novel infectious disease

Abstract

Many zoonotic, novel infectious diseases in humans appear as sporadic infections with spatially and temporally restricted outbreaks, as seen with influenza A(H5N1). Adaptation is often a key factor for successfully establishing sustained human-to-human transmission. Here we use simple mathematical models to describe different adaptation scenarios with particular reference to spatial heterogeneity within the human population. We present analytical expressions for the probability of emergence per introduction, as well as the waiting time to a successful emergence event. Furthermore, we derive general analytical results for the statistical properties of emergence events, including the probability distribution of outbreak sizes. We compare our analytical results with a stochastic model, which has previously been studied computationally. Our results suggest that, for typical connection strengths between communities, spatial heterogeneity has only a weak effect on outbreak size distributions, and on the risk of emergence per introduction. For example, if R₀ = 1.4 or larger, any village connected to a large city by just ten commuters a day is, effectively, just a part of the city when considering the chances of emergence and the outbreak size distribution. We present empirical data on commuting patterns and show that the vast majority of communities for which such data are available are at least this well interconnected. For plausible parameter ranges, the effects of spatial heterogeneity are likely to be dominated by the evolutionary biology of host adaptation. We conclude by discussing implications for surveillance and control of emerging infections.

Figure 1. Schematic representation of the short-term commuting model.

is the number of residents present in the village and the number of commuters in the city. The city has an infinite number of residents. is the per capita commuting rate from the village to the city and the return rate. Together, both determine the number of commuters .

Figure 2. Comparison of the probability of emergence per introduction.

Shown is the probability of emergence as a function of the mutation rate and the intermediate reproductive number . The average reproductive number for each strain is and the mutation rate , . The solid lines are analytical results. The data points represent the average probability of emergence over 1000 simulated emergences with a host population size of . The agreement between analytical calculations and simulations is excellent. Further, the probability of emergence grows non-linearly with and .

Figure 3. Outbreak size distribution.

The solid black line represents the analytical result for an infinite population. Data points are the average probability of simulations. Squares represent the punctuated route (I), and triangles the gradual route (II), both with the mutation probability . A Outbreak size distribution as a function of host population size. For a population size, , of 50, 200 and 500, an emergence event is defined as having at least of the population size infected with the fully adapted strain, while the number was fixed at infected with the fully adapted strain for . While host populations with clearly show finite size effects, even small host populations with can be effectively treated as infinite as the outbreak size is small compared to the population size. The figure reveals the effect of the pathogen's evolution on the outbreak size distribution as the distributions group according to the route of adaptation. B Probability of emergence in the city with short-term commuting. The probability is a function of the overall number of infectious hosts in the village-city model with . The red color represents , blue , green and gold commuters. As for homogeneous populations, the outbreak size distributions group according to the evolutionary route of adaptation. Spatial heterogeneity does not have an influence as all simulations do not show a significant variation from the analytical results.

Figure 4. Probability of emergence in the city with short-term commuting.

The probability is a function of the overall number of infectious hosts in the village-city model. The solid black lines represent the analytical results for an infinite population without spatial heterogeneous structure. Data points are the average probability of simulations. Squares represent the punctuated route (I), and triangles the gradual route (II), both with the mutation probability . As with the outbreak size distribution, probabilities group according to the route of adaptation instead of connection strength in number of commuters. However, the probability of emergence does not converge to for commuters, revealing the effect of spatial heterogeneity when the number of commuters is small relative to the average reproductive number for the well-adapted strain.

Figure 5. Data of commuting patterns in different parts of the world.

Shown is the cumulative fraction of all communities with equal or less than the specified number of commuters. The data was mostly collected by the National Statistical Offices of the respective countries. The gold line represents commuting data from Brazil , the red line data from the USA , the blue line data from the UK , the brown line data from Japan , the cyan line data from Hong Kong , and the green line data from two independent sources. The green line has orange and pink data points , corresponding to its data sources. Our data represents the commuting flows between administrative units. The definition of administrative units varies highly between countries. For example, the US data is on a granularity of counties, while the data from Japan is based on its prefectures. However, heterogeneity can also be found within countries datasets. The Brazilian data is on a level of municipalities with resident sizes ranging from to .

Figure 6. Deviation between simulated and analytical predicted probability of emergence.

The deviation is a function of the average number of commuters . The deviation is defined as . The analytical probability of emergence is for an infinite ulation without spatial structure. The simulated probability of emergence is for short-term commuting with the blue data points representing the gradual route (II) of adaptation, and the orange data points representing the punctuated route (I). The solid black line is , as defined in equation 12 in the main text. It is the analytical expected deviation for spatial heterogeneity as a function of the spatial homogeneity coefficient. The simulations agree very well with the analytical expected deviation. The gradual route (II) is slightly more off from the theoretical prediction as a result of the small but significant number of infected with the intermediate strain. Effectively, it lowers the number of commuters infected with the fully adapted strain and therefore the probability of transmission from the village to the city. Nevertheless, the analytical prediction as well as the simulations show no significant impact of spatial heterogeneity from a critical commuter threshold of .

Figure 7. Impact of spatial heterogeneity on disease transmission between communities (I).

The impact is measured with the spatial homogeneity coefficient with . Given every emergence in the village automatically leads to an emergence in the city, and represents no chance of successfully transmitting the pathogen into the city. The figure reveals that spatial structure becomes especially important for small average reproductive numbers. In addition, the average number of commuters needed to show an effect of spatial heterogeneity is surprisingly small.