Viral evolution and the competitive exclusion principle

Bioscience Hypotheses (2008) 1, 168e171 available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/bihy

Viral evolution and the competitive exclusion principle M.N. Burattini a,b,d, F.A.B. Coutinho a,b,d, E. Massad a,b,c,d,* a

School of Medicine, University of Sa˜o Paulo, Rua Teodoro Sampaio 115, CEP 05405-000, SP, Brazil LIM01eHCFMUSP, SP, Brazil c London School of Hygiene and Tropical Medicine, London University, UK d LIM47eHCFMUSP, SP, Brazil b

Received 15 April 2008; accepted 15 May 2008

KEYWORDS Viral evolution; Competition; Basic reproduction number; Coexistence

Abstract Consider N virus strains. They have to compete within the host (direct competition) and for infecting hosts in a population (indirect competition). Let R0c(i) be the basic reproduction number of strain i within the host (subscript c stands for cell ). Let R0p(i) (subscript p stands for population) be the basic reproduction number of the virus in a host population. We present a simple model that suggests that all the virus strains can coexist only if R0c(1) > R0c(2).. and R0p(1) < R0p(2), that is, for two strains, the winning strain within the host is the loser between hosts. ª 2008 Elsevier Ltd. All rights reserved.

Introduction Viral evolution has been the subject of intensive investigation in the last three decades [1e3]. However, authors have essentially concentrated on the evolution of virulence levels, also called pathogenicity. The question of viral competition and its theoretical and practical consequences has received less attention in the current literature. Assume an infectious agent (a virus) that has N strains. These N strains must compete between themselves in two different environmental levels: within the host; and when they propagate between the hosts (see empirical evidence * Corresponding author. School of Medicine, University of Sa ˜o Paulo, Rua Teodoro Sampaio 115, CEP 05405-000, SP, Brazil. Tel.: þ55 11 3061 7435; fax: þ55 11 3061 7382. E-mail address: [email protected] (E. Massad).

of these two levels in Ref. [4]). The purpose of this paper is, using a very simple model, to study the consequences of the interplay of these two levels of competition on the possibility of the strains to survive. As we shall see each strain, say strain i, is characterized by two basic reproductive numbers [5] that are defined precisely in the main body of this paper. The first basic reproductive number, R0c(i) (the subscript c stands for cells), describes the strain competence within the host. The second basic reproductive number, R0p(i) (the subscript p stands for population), describes the strain competence in propagating themselves between the hosts. We find, with this model, that the only possibility for the N strains coexist is that the following chain of inequalities hold: R0c(1) > R0c(2). > R0c(N ); and R0p(1) < R0p(2). < R0p(N ), that is, that the more competitive strain within the host is the less competent as an infectious agent due to its less adept behavior as an infectious agent between the hosts.

1756-2392/$ - see front matter ª 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.bihy.2008.05.003

Viral evolution

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This paper is organized as follows. In the first section we state our hypothesis. In the next section we consider the competition between the strains, within the host, to invade cells. Then we consider the competition to invade hosts in a population. We consider, in detail, the case of only two strains of virus competition. This is for simplicity only, since the generalization for N strains is straightforward. In addition we present the numerical results for three strains. Finally, we compare the model of virus competition presented in this paper with other models and discuss rapidly some of the experimental evidence available for the type of model described in this paper.

The hypothesis Generally, two or more virus strains coexist in the same host population if the dominant strain within the host is less competent as an infectious agent due to its lower fitness as an infectious agent between the hosts.

Competition within the host (direct competition) In order to elaborate the hypothesis above, we propose a simple model that takes into account the dynamics of viral competition at two levels: within the host (the subject of this section); and between susceptible hosts (subject of the next section). In this section, we consider competition between two strains of the same virus within the host. This is called direct competition in Ref. [4]. Lets consider that the hosts’ body is a population of cells and that the virus infects cells, multiplies and searches for new cells to infect. We consider that the population of cells is kept constant by the functioning of the organism as a whole. Let Sc be the number of susceptible cells. Let Vci(i Z 1,2) be the number of cells infected with strain 1 and with strain 2, respectively. We may write dSc Z  b1c Sc Vc1  b2c Sc Vc2  mSc þ L dt dVc1 Zb1c Sc Vc1  ac1 Vc1  b1c Vc1 Vc2 dt dVc2 Zb2c Sc Vc2  ac2 Vc2  b1c Vc1 Vc2 dt

ð1Þ

Let’s us briefly explain the terms of the above equations. We assume that the cells contaminated say with strain Vc1, sheds b1 virion particles per unit time. A fraction b2 of those virus particle infect susceptible cells Sc. So b1 b2 Vc1 Sc hbic Sc Vc1 new cells became infected per unit time with the strain 1. Those cells die with a rate ac1. We set LZmSc þ ac1 Vc1 þ ac2 Vc2 so that the total population of cells Nc ZSc þ Vc1 þ Vc2 is constant. Superinfection is considered by the terms bicVciVcj and as will be explained in below implies that each cell can be infected by only one strain of virus, dying when there is superinfection. The system of Eq. (1) has three equilibrium points (a) No virus. Sc Z Nc, Vc1 Z 0 and Vc2 Z 0. This equilibrium is a stable attractor if the basic reproductive ratio of both strains, R0c1 Zb1c Nc =ac1 and R0c2 Zb2c Nc =ac2 , are

both less than one. In this case the disease does not maintain itself inside the host. (b) Only strain 1. Sc Z Neq s Nc, Vc1 ZVc1eq Zðm=b1c Þ ðL=ac1 Þ and Vc2 Z 0. In this case strain 2 disappears and only strain 1 survives. This equilibrium is stable if R0c1 Zðb1c Nc =ac1 Þ > R0c2 Zðb2c Nc =ac2 Þ. (c) Only strain 2. Sc Z Neq s Nc, Vc2 ZVc2eq Zðm=b2c Þ ðL=a2c Þ and Vc1 Z 0. In this case strain 1 disappears and only strain 2 survives. This equilibrium is stable if R0c1 Zðb1c Nc =ac1 Þ > R0c2 Zðb2c Nc =ac2 Þ. The result above is called in the literature as the competitive exclusion principle [6]. It is easy to understand the results intuitively by focusing in the equations for Vc1 and Vc2. Let’s first consider the equations without the reinfection terms dVc1 Zb1c Sc Vc1  ac1 Vc1 dt dVc2 Zb2c Sc Vc2  ac2 Vc2 dt

ð2Þ

It is easy to see that the virus strain V1 decreases when b1c Sc  ac1 < 0 and that strain V2 decreases if b2c Sc  ac2 < 0. Therefore, an equilibrium with both Vc1eq and Vc2eq can only occur for a value of Sceq that is equal to both b1c/ac1 and b2c/ac2, which is impossible. The superinfection terms, b1cVc1Vc2 and b1cVc1Vc2 do not change the possible equilibrium solutions of the system and cannot change their stability properties. In fact they decrease the value of Sceq and hence we can only have the equilibrium when one strain is excluded. Alternative models and mechanisms for viral competition inside the host body will be discussed below.

Competition between the hosts (indirect competition) We have seen above that inside each host one can have only one strain. Now we shall consider a simple model for the competition between the hosts. (This competition is called indirect competition [4].) Without loosing generality we may assume that the dominant strain inside the host is strain 1. Thus if a host is infected with strain 2 and catches strain 1 this strain will replace strain 2 in a period of time that we will assume to be very short in comparison to the time involved in the population transitions. On the other hand if a host is infected with strain 1 and catches strain 2 nothing happens, that is, strain 2 is unable to dislodge strain 1. We then write for the population dynamics the following system dS Z  b1p SV1 =N  b2p SV2 =N  mS þ L dt dV1 Zb1p SV1 =N  ap1 V1 þ b2/1 V2 V1 =N dt dV2 Zb2p SV2 =N  ap2 V2  b2/1 V2 V1 =N dt

ð3Þ

In this system S, V1 and V2 are the number of susceptible hosts, hosts infected with strain 1 and hosts infected with strain 2, respectively, and NZS þ V1 þ V2 is the total population. The parameter bpi is the product of the number

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of contacts a host makes with other host beings per unit time, times the probability of getting strain i if the contact is made with a host infected with this strain, which occurs with probability Vi/N. Note that if a host infected with strain 2 makes contact with a host infected with strain 1 he/she becomes a host infected with virus 1with rate b2/1 V2 V1 =N. As explained above this occurs because strain 1 rapidly displaces strain 2 from the host. The parameters api are the mortality rates of hosts infected with strain i, m is the death rate of the susceptible individuals and L is the influx of susceptible. This system has four equilibrium points: (a) No disease. The equilibrium are S Z L/m, V1 Z 0 and V2 Z 0 representing the case were there is no disease. This equilibrium is stable if the individual basic reproductive number, R0p1 Z b1/a1 and R0p2 Z b2/a2 are both <1. (b) Only strain 1. This occurs when R0p1 > R0p2 or if R0p1 < R0p2 but b2/1 is big enough. (c) Only strain 2. This occurs when R0p1 < R0p2 and b2/1 are not big enough. (d) Coexistence. The viruses’ strains can coexist only if R0c(1) > R0c(2) and R0p(1) < R0p(2). The equilibrium points in this case are b2/1 L Seq Z b2/1 m  b1 a2 þ b2 a1 NZSeq

For those parameters’ values the resulting basic reproductive numbers are: R0p1 Z 1.01, R0p2 Z 2.63, and R0p3 Z 8.84. Table below shows the resulting equilibrium prevalence of the three strains with three different sets of transition parameters bij. The values of the parameters, in arbitrary units, and the equilibrium proportions are: V1

V2

V3

3

b2/1 Z 2  10 b3/1 Z 2  103 b3/2 Z 2  103

0.0167

0.7852

0.1981

b2/1 Z 2  103 b3/1 Z 2  103 b3/2 Z 1  104

0.0111

0.000

0.9889

b2/1 Z 2  103 b3/1 Z 1  104 b3/2 Z 1  104

0.000

0.000

1.000

The results show that, by choosing the transition parameters bij all possible equilibrium outcomes can be obtained.

Other mechanisms of viral competition ð4Þ

b2/1 þ b2  b1 b2/1 þ a2  a1

ð5Þ

b Seq  a2 N V1eq Z 2 b2/1

ð6Þ

a1 N  b1 Seq V2eq Z b2/1

ð7Þ

The existence of these four equilibrium (4e7) establishes our result. Below we show the model equations and the numerical results for three strains. The model equations are: dS Z  b1p SV1 =N  b2p SV2 =N  b3p SV3 =N  mS þ L dt dV1 Zb1p SV1 =N  ap1 V1 þ b2/1 V2 V1 =N þ b3/1 V3 V1 =N dt dV2 Zb2p SV2 =N  ap2 V2  b2/1 V2 V1 =N þ b3/2 V3 V2 =N dt dV3 Zb3p SV3 =N  ap3 V3  b3/1 V3 V1 =N  b3/2 V3 V2 =N dt

ð8Þ

We tested the model using the following set of parameters, arbitrarily chosen in arbitrary units. They are as follows: Parameter

Value

b1p b2p b3p ap1 ap2 ap3 m L

2.02  103 2.50  103 7.10  103 2.00  103 1.00  103 8.00  104 1.00  104 0.14321

The mechanism of direct competition considered in this paper is a competition for cells. It relies on the hypothesis that if a cell is infected by two strains in succession then the second infection immediately kills the cell. This is implied in the model described in Section 2 where the superinfection terms, b1cVc1Vc2 and b1cVc1Vc2 immediately kill the cell. If a cell is not killed by superinfection then the two strains can coexist inside the body. In this case the Eq. (1) should be replaced by dSc Z  b1c Sc Vc1  b2c Sc Vc2  mSc þ L dt dVc1 Zb1c ðSc þ Vc2 ÞVc1  ac1 Vc1 dt dVc2 Zb2c ðSc þ Vc1 ÞVc2  ac2 Vc2 dt

ð9Þ

It is easy to check that in this case the competitive exclusion principle does not necessarily hold and the two strains can indeed coexist. Other mechanisms where proposed for pathogen competition. The papers by Massad et al. [7] and by Cheon [8] propose a LotkaeVolterra model for the competition between different strains. In Ref. [9], the competitive exclusion principle is demonstrated for HIV strains competing for host (indirect competition). In Ref. [10], it is introduced a hyperparasite that competes with the pure strain and deduce conditions for the hyperparasite to win. In this model, the hyperparasite infects the cells with the pure strain, and replaces it without killing the cells. In vitro studies of competition between different subtypes of HIV [11] demonstrate a clear exclusion pattern in the sense that in a cell culture infected with two HIV subtypes, one of them will be rapidly excluded by the other. Finally, we would like to comment that, except for the paper by Cheon [8] all the models discussed predict that

Viral evolution coexistence within the hosts is short-lived phenomenon not feasible in the long run [12].

Conclusions To the best of our knowledge this is the first model to consider viral competition at two levels, inside the hosts and among hosts, although empirical evidence for competition at the host and at a vector levels was reported by Power [4] for plant viruses. The basic mechanisms described in this work can be applied this much more complicated system, involving two virus strains, two vectors and one definitive host. In conclusion, coexistence of two or more competing strains is possible whenever R0c(1) > R0c(2). > Roc(N ); and R0p(1) < R0p(2). < Rop(N ), that is, that the more competitive strain within the host is the less competent as an infectious agent due to its less adept behavior as an infectious agent between the hosts.

Conflict of interest The authors declare that there is no conflict of interest related to this manuscript.

Acknowledgements This work was supported by FAPESP, CNPq and LIM01e HCFMUSP.

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