
Restif and Grenfell (2006), Integrating life history and crossimmunity into the evolutionary dynamics of pathogens.
November 2010, model of the month by Lukas Endler
Original model: BIOMD0000000249
The dynamics of the spreading of infectious diseases in populations have been shown to be well describable by mathematical modelling. Classical models have helped to understand and predict the development of various epidemics, such as AIDS and influenza, and given hints on possible courses of action for containment and prevention [1,2]. Important points in these models, that have been addressed increasingly in recent years, are the existence and emergence of new variants and strains of specific pathogens and their evolution. Various approaches have been developed, some more focused on the evolutionary aspects of virulence [3], others on the evolution of immunity of hosts against and crossimmunity between related strains of a pathogen [4].
Virulence describes the tendency of a pathogen to be harmful for a host. The tradeoff hypothesis, a widely accepted theory of virulence evolution, states, that to counteract the high host mortality of increased virulence, it has to go along with increased transmission ability [3].
Another important factor influencing pathogen evolution lies in the immune response of the host. After infection by one strain, the host's acquired immunity can also decrease susceptibility to another, related strain, a phenomenon called crossimmunity.

Figure 1: Schematic representation of a simple SIR model. Expressions above the arrows indicate net per capita flows, dotted arrows stand for birth and death.

Figure 2:
Schematic representation of the extended SIR model. The expressions next to the connecting arrows indicate per capita flows between compartments. The net flux can be obtained by multiplying the per capita flows with the amount of individuals in the source compartment. The birth and death flows are or included. The force of infection of strain i: Λi = βi * (Ii +Ii')/N.
Figure taken from [5].

While both of these factors, the evolution of virulence and the occurrence of crossimmunity, are of great importance for the development of pathogens and epidemics, they rarely are modelled in combination, but rather studied separately. In their article, Restif and Grenfell [5] developed a novel approach to explore the implications of both in one framework. The authors analysed the invasion of one pathogen strain into a population, in which another strain is already endemic and established, under consideration of virulence and crossimmunity. They chose whooping cough as an example, although their approach is generally applicable.
Whooping cough is caused by the bacterium Bordetella pertussis, and yearly affects nearly 50 Mio people causing more around 300 000 deaths. While vaccines exist, they have been derived in the 1950s, and new strains, against which existing vaccines have been shown to be less protective, have emerged since [6,7]. This makes the interplay of new strains an especially pressing problem.
For their approach, the authors modified a classical SusceptibleInfectiousRecovered (SIR,) framework. In this kind of models, a population is divided into different nonoverlapping subpopulations called compartments. The compartments stand for the groups of uninfected, susceptible individuals, S, the infected ones, I, and the ones that have recovered and have acquired immunity to infection, R (figure 1). Flow of individuals from one compartment to the next is determined by certain key parameters and most often using simple Mass action kinetics. New susceptible individuals arrive with a birth rate balancing the overall amount of deaths, with the death rate, μ, the inverse of the average life expectancy, being equal for all compartments. The number of infections is mainly determined by the transmission rate, β, and the recovery rate, γ, which again is the inverse of the average infectious period. The ratio of transmission to recovery rate, R0 = β/ γ, is known as the basic reproduction number, a characteristic of the pathogen. After recovery from the disease, individuals stay immune for a certain time period on average that is inverse to the loss of immunity a rate, σ.

The authors extended the standard SIR model to accommodate two pathogens, strain 1 and 2, by doubling the amount of infected and resistant compartments. To include crossimmunity three additional compartments had to be added: I1' and I2' that indicate individuals resistant to strain 1 infected with strain 2 and vice versa, and R', standing for individuals that acquired immunity against both strains in sequence (see figure 2). The transmission and recovery rates, β1/2 and γ 1/2 , were assumed to be strain specific, but their basic reproductive ratio, R0, was set equal. This was done to allow for a tradeoff between and aggressive and prudential pathogen behaviour, with high transmission rates, but short infectious periods of vice versa [8].
Crossimmunity, φ, on the other hand, was taken to be symmetrical and strain independent. φ stands for the partial immunity to infection with one strain, that is granted by immunity to the other strain. For example, for individuals immune to strain 1, a value of φ = 0.1 ten percent protection against infection with strain 2, a value of φ = 0.5 means a halving of the infection rate, while φ = 1 stands for full immunity.
To elucidate the interplay between crossimmunity and strain specific characteristics, the authors varied the transmission and recovery rates of the invading strain 2, β2 and γ2, at varying levels of crossimmunity, φ. As initial conditions, a population stably infected with strain 1 was used with an addition of a small number of individuals infected with strain 2. The main interest of the authors was to see under which conditions one strain could outcompete the other leading to potential extinction of that strain, and which parameter ranges allowed coexistence.
For this, the authors first performed a classical deterministic analysis of the system. In the deterministic, continuous case, extinction only occurs as a limit case as one subpopulation asymptotically approaches zero, and only for vastly differing R0 values. For identical R0 values for both strains, the deterministic solution gives either coexistence, or, for very low transmission rates, extinction of both strains. The dynamics for the coexisting cases show an interesting behaviour in dependence of the crossimmunity (figure 3). For strong crossimmunity (figure 3b), strain 2 introduction leads to a deep trough in the amount of individuals infected by strain 1, with a shallower trough in strain 2 infection levels following a few year later, when strain 1 reinvades. The depths of the initial troughs depends, to a big extent, on the level of crossimmunity, but also on the transmission rate of the invading strain. A high transmission rate and corresponding short infectious period leads to deeper troughs of strain 2 after the reinvasion of strain 1. This effect is further enhanced by shorter immune periods (see figure 3 of [5]).

Figure 3: Deterministic solutions of the system after invasion of novel strain (strain 2) into a population with an endemic infection by strain 1, with weak (a : φ = 0.2) and strong (b: φ = 0.8) crossimmunity. Immune period: 1/σ = 20 years, R0 = 17, infectious period: 1/γ = 21 days, life expectancy: 1/μ = 50 years. (after figure 2 of [5])

Figure 4:
Distributions of outcomes of stochastic simulations for varying values of crossimmunity,φ (rows) and immune period, 1/σ (columns), and the infectious period, 1/γ of strain 2. All other parameter values where chosen as in Figure 3. Population size 106, 1000 simulations. (taken from [5])

In a stochastic, discrete interpretation of the model, extinction can occur much more readily. First, strain 2 can get extinct in the initial phase of low infection with that strain. Then, in a later phase, deep troughs can additionally lead to extinction of the other strain. To predict the probabilities of extinction in the stochastic system, the authors tried to use the dynamics of the deterministic solution. They looked at the dependence of the initial growth rates of the invading strain on different parameters, and evaluated the influence of parameters on regions on the initial troughs. For the initial period three factors were predicted to be beneficial for strain 2 survival. It transmission rate should be much higher than that of strain1, crossimmunity should be weak, and the immune period long. For the success following invasion, on the other hand, strong crossimmunity causes deep troughs, allowing for possible trough extinction of strain 1. Additionally a longer infection period of strain 2, which for identical R0, means lower transmission rates, leads to shallower troughs for strain 2 when strain 1 bounces back. This means that strain 2 has tradeoffs between survival in the initial period and the latter one.
The possible outcomes for the stochastic simulations were classified into coexistence, extinction of strain 2 in the initial period, extinction of strain 2 in the trough after strain 1 reinvasion, replacement of strain 1 by strain 2, and total extinction of both strains. Figure 4 shows the outcomes of stochastic simulations for different parameter combinations. As predicted strong crossimmunity influences initial extinction of strain 2, while the effect of the infectious period was contrary to the derivations from the deterministic system concerning survival of the initial period. For the long term development, the immune period proofed to be of big importance. Long immune periods lead to extinction of at least one strain, and in combination with high crossimmunity, even to extinction of both. As predicted, strong crossimmunity was required for strain 2 to replace strain 1, while low crossimmunity favours trough extinction of strain 2. Apart from the joined characteristics of immunity, the strain 2 specific values of transmission rate and infectious period did turn out to influence the outcomes only for certain combinations of medium to long immune periods and high crossimmunity. Again the occurrence of trough extinction of strain 2 and strain 1 follows the predictions, albeit only on a narrower range of immune periods and crossimmunity.
While being relevant for the system the authors in their elegant study show the importance of analysing a epidemiological systems in different ways, both deterministically and in a stochastic framework. The predictions of the deterministic system were qualitatively correct in most cases, but the detailed parameter ranges for which they were valid could only be derived by explicit stochastic simulation, of course also sensitive to the population size.

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