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Modeling Competition Between Two Influenza Strains

  • Rinaldo B. SchinaziEmail author
Conference paper
  • 786 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 38)

Abstract

We use spatial and nonspatial models to argue that competition alone may explain why two influenza strains do not usually coexist during a given flu season. The more virulent strain is likely to crowd out the less virulent one. This can be seen as a consequence of the Exclusion Principle of Ecology. We exhibit, however, a spatial model for which coexistence is possible.

Keywords

Competition models Stochastic process Influenza Swine strain Exclusion principle Ecology 

1 Introduction

The seasonal flu strain was a lot less prevalent during the 2009/2010 influenza season than during the previous years, see Fluview (the weekly CDC influenza report). On the other hand, some time during spring 2009, the new so-called swine strain appeared. There seems to be a relation between these two events. In this chapter we propose to explain this phenomenon using competition models. We will use spatial and nonspatial models to show that in a given flu season coexistence of two strains is unlikely due to competition alone. We will also show that geometry and space may be critical for coexistence. Our models deal with competition over only one flu season. In the real world, because of mutations the fight between two strains may go on for several flu seasons before one strain outcompetes the other. This picture is consistent with the very skinny shape of the phylogenetic tree for influenza; see, for instance Koelle et al. [5] and van Nimwegen [8]. In this chapter, the two competing strains are assumed not to undergo mutations, and therefore the time scale we focus on is one flu season.

A competing explanation of the non-coexistence of the two influenza strains is cross immunity. For instance, immunity may explain why older generations have not been as much affected as the younger ones in the swine flu epidemic. It may be due to some previous exposure to a similar strain, see the discussion in Greenbaum et al.[3]. However, using a cross-immunity argument to explain why the swine strain crowds out the seasonal one may be more difficult. The hypothesis would be that the swine strain must confer some immunity against the seasonal flu. But, clearly the seasonal strain does not confer any immunity against the swine strain: after all even young people (the group most severely affected by the swine strain) have usually been exposed to the seasonal strain and do not seem to be protected against the swine strain. Hence, for this argument to work the swine strain must confer some immunity against the seasonal strain, but the seasonal strain cannot confer any immunity against the swine strain. In contrast to this cross immunity hypothesis we argue in this chapter that even in models for which there is no immunity at all (every individual that recovers is immediately susceptible again!), coexistence of two competing strains is rather unlikely.

2 The ODE Model

Our first model is a system of ordinary differential equations. Let u 1(t) and u 2(t) be the density of individuals infected at time t with strains 1 and 2, respectively. We set
$$\begin{array}{rlrlrl} u_{1}^{\prime} = &\lambda _{1}u_{1}u_{0} - \delta _{1}u_{1} & & \\ u_{2}^{\prime} = &\lambda _{2}u_{2}u_{0} - \delta _{2}u_{2} & & \end{array}$$
where u 0(t) is the density of susceptible individuals at time t. In words, individuals infected with strain i infect susceptible individuals at rate λ i and get healthy at rate δ i , for i = 1, 2. We assume that the only possible states are 0, 1 and 2. Hence, at any time t > 0 we have \(u_{0}(t) + u_{1}(t) + u_{2}(t) = 1\). In particular, as soon as an infected individual gets healthy, it is back in the susceptible pool.
Let 1 be the seasonal and 2 be the swine strains. Some reports indicate that the swine strain may be more virulent than the seasonal strain, see Fraser et al. [2]. Under that assumption,
$$\frac{\lambda _{1}} {\delta _{1}} < \frac{\lambda _{2}} {\delta _{2}}.$$
Assume also that at some point in time the ODE model is at the equilibrium \((0,1 - \frac{\delta _{2}} {\lambda _{2}} )\). That is, there is no seasonal strain and the swine strain is in equilibrium. Now introduce a little bit of seasonal strain (small u 1). Will the seasonal strain be able to grow? Using that u 1 is almost 0 and that u 2 is almost \(1 - \frac{\delta _{2}} {\lambda _{2}}\) we make the approximation
$$u_{0} = 1 - u_{1} - u_{2} \sim1 -\left (1 - \frac{\delta _{2}} {\lambda _{2}}\right ) = \frac{\delta _{2}} {\lambda _{2}}.$$
Hence,
$$u_{1}^{\prime} \sim\lambda _{1}u_{1} \frac{\delta _{2}} {\lambda _{2}} - \delta _{1}u_{1} = u_{1}\left (\lambda _{1} \frac{\delta _{2}} {\lambda _{2}} - \delta _{1}\right ).$$
Since we are assuming that \(\frac{\lambda _{1}} {\delta _{1}} < \frac{\lambda _{2}} {\delta _{2}}\) we get u′ 1 < 0. That is, under these assumptions and according to this model, the seasonal flu will not take hold.

In fact this system of ODE is a particular case of a well-known competition model. For the general version of this model, it is known that one of the strains will vanish; see Exercise 3.3.5 in Hofbauer and Sigmund [4]. The point is that we have two populations (the population of individuals infected with strain 1 and the population of individuals infected with strain 2) that compete for a single resource (the susceptible individuals). It turns out that in such a model, one population will drive the other one out. This is a particular case of the so-called “Exclusion Principle” of Ecology: if the number of populations is larger than the number of resources all the populations cannot subsist in the long run, see 5.4 in Hofbauer and Sigmund [4].

3 The Spatial Stochastic Model

In the preceding model there is no space structure, and all the individuals in the population can be seen as neighbors of each other. In this section, we go to the other extreme where there is a rigid space structure and each individual has a fixed number of neighbors.

We now describe a spatial stochastic model called the multitype contact process, see Neuhauser [7]. Let S be the integer lattice Z d (d is the dimension) or the homogeneous tree T d for which each site has d + 1 neighbors. The system is described by a configuration ξ ∈ { 0, 1, 2} S , where ξ(x) = 0 means that site x is occupied by a susceptible individual, ξ(x) = 1 means that x is occupied by an individual infected by strain 1 and ξ(x) = 2 means that x is occupied by an individual infected by strain 2. If S is Z d , then each site has 2d neighbors, if S is T d , then each site has d + 1 neighbors. For x ∈ S and ξ ∈ { 0, 1, 2} S , let n 1(x, ξ) and n 2(x, ξ) denote the number of neighbors of x that are infected by strain 1 and strain 2, respectively.

The multitype contact process ξ t with birth rates λ1, λ2 makes transitions at x when the configuration of the process is ξ
$$\begin{array}{rlrlrl}1 \rightarrow 0& \mbox{ at rate 1 }& & \cr 2 \rightarrow 0& \mbox{ at rate 1 }& \cr 0 \rightarrow 1& \mbox{ at rate }\lambda _{1}n_{1}(x,\xi ),& \cr 0 \rightarrow 2& \mbox{ at rate }\lambda _{2}n_{2}(x,\xi ),& \cr \end{array}$$
In words, a susceptible individual gets infected by an infected neighbor at rates λ1 or λ2, depending on which strain the neighbor is infected with. An infected individual gets healthy (and is immediately susceptible again) at rate 1. Note that compared to the ODE model, we are assuming in this model that \(\delta _{1} = \delta _{2} = 1\). This is so because most of the mathematical results have been proved under the assumption δ1 = δ2. We take this common value to be 1 to minimize the number of parameters.
The multitype contact process is a generalization of the basic contact process which has only one type. The transitions of the basic contact process are given by
$$\begin{array}{rlrlrl}1 \rightarrow 0& \mbox{ at rate 1 }& & \cr 0 \rightarrow 1& \mbox{ at rate }\lambda _{1}n_{1}(x,\xi ),& \cr \end{array}$$
For the basic contact process, there exists a critical value λ c whose exact value is not known and which depends on the graph the model is on. If λ1 > λ c , then starting with even a single infected individual, there is a positive probability of having infected individuals at all times somewhere in the graph. On the other hand, if λ1 ≤ λ c , then starting from any finite number of infected individuals all the infected individuals will disappear after a finite time. See Liggett [6] for more on the basic contact process on the square lattice and on trees.

The Space Is the Square Lattice Z d

We now go back to the multitype contact process. Assume that λ2 > λ c and λ2 > λ1 then there is no coexistence of strains 1 and 2 in the sense that

$$\lim\limits_{t\rightarrow\infty }P(\xi _{t}(x) = 1,\xi _{t}(y) = 2) = 0$$
for any sites x and y in Z d and any initial configuration ξ0. In fact, strain 2 always drives out strain 1 in the following sense. Let A be the event that strain 2 will not ever disappear. Then,
$$\lim\limits_{t\rightarrow\infty }P(\xi _{t}(x) = 1\vert A) = 0,$$
for any site x in Z d and any initial configuration (see Theorem 2 in Cox and Schinazi [1] and also Neuhauser [7]). Hence, assuming that λ2 > λ1 (i.e., strain 2 is more virulent than strain 1) this model too predicts that the seasonal flu will be crowded out by the swine strain. The spatial structure seems to have no influence on the outcome. The next section will show that this is not always so and that a different (more crowded) space structure allows coexistence.

The Space Is the Tree T d

There is a fundamental difference between the basic contact process on the square lattice and the same model on the tree. There are two (instead of one) critical values for the basic contact process on the tree. The definition of λ c is as before. We also define another critical value λ cc in the following way. Consider the basic (one type) contact process with birthrate λ1. Let O be a fixed site on the tree or square lattice. Start the process with a single infected individual at O. The probability that the infection will return to site O infinitely many times is positive if and only if λ1 > λ cc . It turns out that λ c  < λ cc on the tree but λ c  = λ cc on the square lattice.

The fact that the basic contact process has two distinct critical values on the tree has interesting consequences for the multitype contact process on the tree. Let λ1 and λ2 be in (λ c , λ cc ), and then strains 1 and 2 may coexist on the tree in the following sense. Under suitable initial configurations we have for any sites x and y
$$\mathop{\lim\inf}\limits_{t\rightarrow\infty }P(\xi _{t}(x) = 1,\xi _{t}(y) = 2) > 0.$$
See Theorem 1 in Cox and Schinazi [1]. Note that coexistence occurs even for λ1 < λ2 but both parameters need to be in the rather narrow interval (λ c , λ cc ). This result shows that space structure and geometry may be crucial in allowing coexistence.

4 Discussion

Our models show that at least in theory coexistence of two competing strains is unlikely. Coexistence is however possible for the multitype contact process on a tree. The tree can be thought of as a model for high-density populations (in a ball of radius r there are \((d + 1){d}^{r-1}\) individuals on the tree T d but only about r d on the lattice Z d ). In order to have coexistence, both infection rates cannot be too low or too high but may be unequal. In all other cases, there will be no coexistence on the tree, and there is never coexistence on Z d unless λ1 is exactly equal to λ2, a rather unlikely possibility, see Neuhauser [7]. Interestingly, the behavior of the mean-field ODE model is the same as the behavior of the model on Z d but different from the model on the tree. In general, it is expected that the model on the tree to be closer to the mean-field model than to the model on Z d . This is not so in this example.

References

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of Colorado at Colorado SpringsColorado SpringsUSA

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