Insights into the Evolution and Emergence of a Novel Infectious Disease

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 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.

with φ i (j i , . . . , j n ) being the probability that a single infected host of type i gives rise to j k secondary infections with k ∈ {i, . . . , n}. Hence, for a geometric offspring distribution governed by (2) in the main text, φ is φ n (j n ) = p jn n r n φ n−1 (j n−1 , j n ) = p jn−1 n−1 r n−1 + p jn−1 n−1 m n−1 p jn n r n . . . The probability of extinction q i is defined as probability of an introduction which will reach only a finite outbreak size, and go extinct. Hence it is the probability that all secondary cases of every infected host finally lead to extinction. This self-similarity is expressed with It is possible to calculate all extinction probabilities, starting with the fully adapted strain. Inserting the probabilities of (2) and taking into account that p i , q i < 1, the pgfs in (1) become, under the use of the fixed point relation (3), It is possible to find a closed form for the probabilities of extinction with a few lines of derivations. If If p i > 0, equation (4) becomes Note that q i with p i > 0 has two analytical solutions with q i = 1 as trivial one. Only the solution shown in (6) needs to be explicitly calculated.
If a wildtype strain is introduced at each introduction, the probability of extinction becomes q 0 . The probability of emergence per introduction p em is one minus the probability of extinction This expression can be solved analytically for all possible routes of adaptation.

A.1.2 Outbreak Size Distribution
An emerging disease in a population can be modelled as individual particles with new offspring, as done in the branching process calculations above. State-space models are another representation. These focus not on individual hosts, but the "states" in which these hosts are. Instead of having X hosts moving between the states of being susceptible, infected, or recovered, the system gets transformed into moving through Y states corresponding to the number of susceptibles, infected, or recovered hosts. The probabilities to infect, mutate, or recover are constant over time for each host, and so are the transition rates between the states.
Let the system be in state I i,t , corresponding to the number of infectious hosts with R i at transition t. The number of transitions t is similar to a time as it goes only in one direction. Each change of state, which means every infection, mutation, or recovery, is a single transition. Hence the rate of entering the state I i,t+1 + 1 is equal to the probability of infection plus the probability of mutation, p i and m i−1 . The transition into state I i,t+1 − 1 can happen either by mutation or recovery. Hence the according transition rates are m i and r i . This leads to the following Master-Equation And for I i,t = 0 The initial condition at t = 0 is all probabilities are 0 except ρ i (I i,0 , 0) = 1. I i,0 is the number of infected hosts to start with, also known as "patient zeros". Going back to the branching process interpretation, each infected host is assumed independent from each other. The same idea is true in the Master-Equation interpretation. The time of the transition through mutation is irrelevant, only the transition event itself counts. Hence it is possible to sum up all the mutation events from strain i − 1 to i and add these number to the initial number of infected hosts with strain i at t = 0.
In addition, the end condition is also known. Let strain i go extinct at transition t = T . Hence the system must have been in state 1 at transition t = T − 1 and of the past transition events must have been infections. x i is also the outbreak size of secondary infected hosts. Accordingly the number of transitions for a given outbreak size is Equation (12) reveals a direct relation between the number of transitions and the number of secondary infections. It enables us to derive the outbreak size probability g i (I i,0 , x i ), as it is possible to derive ρ i (1, t) for every t with the initial condition I i,0 and the Master-Equations (8), (9), and (10). In general, the outbreak size probability is Note that x i is only the number of new infected hosts in i, not the mutated ones I i,0 . It is necessary to subtract these to prevent over-counting of infected. The derivations can be done analogous to the ones The number of initially infected hosts I i,0 depends on x i−1 and the I i−1,0 as only infected host with strain i − 1 can possibly mutate into strain i. The probability of mutation for each infected host is µi−1 µi−1+1 = mi−1 mi−1+ri−1 . As the number of mutations is binomial distributed, the probability of this initial condition is It is possible to calculate the distribution of each strain by recursive derivation, starting with the wildtype strain. Furthermore the overall outbreak size probability can be derived using conditional probabilities p out (X) = g 1 (I 1,0 , x 1 ) I1,0+x1,...,In−1,0+xn−1 I2,0,...,In,0=0 with X = I 1,0 + n i=1 x i as overall outbreak size. In addition, it is straight forward to derive the probability of emergence given p out While equation (7) is the fastest way to compute p em , equation (17) can be used as well.

A.1.3 Waiting Time to Emergence
Let the probability of emergence per introduction be p em with 0 ≤ p em ≤ 1. Hence the probability of having an emergence after M introductions is with q ext = 1 − p em as probability of extinction per introduction. A convenient way to handle probability distributions is with pgfs with |z| < 1 qext . Using known characteristic of pgfs, it is possible to calculate the expected number of introductions Note that this is the average number of introductions without an emergence, as defined in (18). The average number of introductions needed to get an emergence event is M + 1.
In addition, the variance can be obtained in a similar way The variance is very large, and therefore the peak around the expected value very wide. The standard deviation is of the same magnitude as the mean. This makes the number of needed introductions highly unpredictable.

A.2 Spatial Heterogeneity on Larger Timescales
As for commuting on a daily basis, we focus on a simple village -city model to approximate the spatial heterogeneity in rural areas only weakly connected to bigger cities. As before, we have a wildtype pathogen capable of acquiring adaptations for human transmission. Assume N (v) = 1000 villagers, and an infinite number of hosts in the city. Moreover, assume a commuting rate of χ between city and village.
As discussed in the main text, we assume here that commuting is on a timescale longer than the infectious period, but may be temporary. Thus we assume that the return rate is the same as the outgoing rate, in order to maintain a constant number of hosts in the village. Figure 1 shows a schematic representation of the model. is the number infected with type i in the city. Then, assuming that all strains have the same mean infectious period, and normalising continuous time with respect to this mean infectious period, an ODE model for this system reads where µ i and α i are, as before, respectively the mutation number and basic reproductive number associated with strain i. Our initial conditions are I   Nevertheless, this is an very unlikely scenario with a probability of less than 0.02 at 100 infected hosts.
Interestingly, five infectious hosts or more are need in the gradual route (II) before emergence becomes more likely than extinction (see Figure 3). In the punctuated route (I), this becomes three or more infected, caused by the very low probability of generating secondary infected with an average reproductive number of α 2 = 0.1. Secondary infections are most likely to occur with the fully adapted strain only. . Probability of emergence in the city with long-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 10 7 simulations. Squares represent the punctuated route (I), and triangles the gradual route (II), both with the mutation probability µ = 0.1 (see main article). The orange colour represent an average number of commuters c = 10, the blue colour c = 50, and the red colour c = 100. The plot is dived at 30 infectious hosts into a linear probability scale (left), and a log10 probability scale (right). 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 1 for c = 10 commuters. Here the effect of spatial heterogeneity is very small but visible as a small number of commuters can lead to an emergence in the village without further, sustained transmission to and within the city.