Stochastic Modeling and Simulation of Viral Evolution

Abstract

RNA viruses comprise vast populations of closely related, but highly genetically diverse, entities known as quasispecies. Understanding the mechanisms by which this extreme diversity is generated and maintained is fundamental when approaching viral persistence and pathobiology in infected hosts. In this paper, we access quasispecies theory through a mathematical model based on the theory of multitype branching processes, to better understand the roles of mechanisms resulting in viral diversity, persistence and extinction. We accomplish this understanding by a combination of computational simulations and the theoretical analysis of the model. In order to perform the simulations, we have implemented the mathematical model into a computational platform capable of running simulations and presenting the results in a graphical format in real time. Among other things, we show that the establishment of virus populations may display four distinct regimes from its introduction into new hosts until achieving equilibrium or undergoing extinction. Also, we were able to simulate different fitness distributions representing distinct environments within a host which could either be favorable or hostile to the viral success. We addressed the most used mechanisms for explaining the extinction of RNA virus populations called lethal mutagenesis and mutational meltdown. We were able to demonstrate a correspondence between these two mechanisms implying the existence of a unifying principle leading to the extinction of RNA viruses.

Appendices

Appendices

A Review of Multitype Branching Process Theory

A discrete-time multitype branching process with types or classes indexed by a nonnegative integer r ranging from 0 to R is described by a sequence of vector-valued random variables $\varvec{Z}_n=(Z_n^0,\ldots ,Z_n^R)$, ($n=0,1,\ldots$), where $Z_n^r$ is the number of particles of type or class r in the nth generation. The initial population is represented by a vector of nonnegative integers $\varvec{Z}_0$ (also called a multi-index) which is nonzero and non-random. The time evolution of the population is determined by a vector-valued discrete probability distribution $\varvec{\zeta }(\varvec{i})=\big (\zeta _r(\varvec{i})\big )$, defined on the set of multi-indices $\varvec{i}=(i^0,\ldots ,i^R)$, called the offspring distribution of the process, which is usually encoded as the coefficients of a vector-valued multivariate power series $\varvec{f}(\varvec{z})=\big (f_r(\varvec{z})\big )$, called probability generating function (PGF).

The mean matrix or the matrix of first moments$\varvec{M}=\{M_{ij}\}$ of a multitype branching process describes how the average number of particles in each type or class evolves in time and is defined by $M_{ij}=\mathbf {E}(Z_1^i|Z^j_0=1)$, where $Z^j_0=1$ is the abbreviation of $\varvec{Z}_0=(0,\ldots ,1,\ldots ,1)$. In terms of the probability generating function $\varvec{f}=(f_0,\ldots ,f_R)$ it is given by

$$\begin{aligned} M_{ij}=\dfrac{\partial f_j}{\partial z_i}(\varvec{s})\bigg |_{\varvec{s}=\varvec{1}} \end{aligned}$$

(3)

where $\varvec{1}=(1,1,\ldots ,1)$. Typically, the mean matrix $\varvec{M}$ is nonnegative, and hence it has a largest nonnegative eigenvalue. When the largest eigenvalue is positive, it coincides with the spectral radius of $\varvec{M}$ and it is called, following Kimmel and Axelrod (2002), the malthusian parameter$\mu$.

The vector of extinction probabilities of a multitype branching process, denoted by $\varvec{\gamma }=(\gamma _0,\ldots ,\gamma _R)$, where $0 \leqslant \gamma _r\leqslant 1$, is defined by the condition that $\gamma _r$ is the probability that the process eventually become extinct given that initially there was exactly one particle of class r.

The classification theorem of multitype branching processes states that there are only three possible regimes for a multitype branching process (Harris 1963; Athreya and Ney 1972; Kimmel and Axelrod 2002):

Super-critical::

If $\mu >1$ then $0\leqslant \gamma _r<1$ for all r and, with positive probability the population survives indefinitely.

Sub-critical::

If $\mu <1$ then $\gamma _r=1$ for all r and with probability 1 the population becomes extinct in finite time.

Critical::

If $\mu =1$ then $\gamma _r=1$ for all r and with probability 1 the population becomes extinct; however, the expected time to the extinction is infinite.

When a multitype branching process is super-critical, it is expected that, according to the “Malthusian Law of Growth” it will grow indefinitely at a geometric rate proportional to $\mu ^n$, where $\mu$ is the malthusian parameter, $\varvec{Z}_n \approx \mu ^n \,\varvec{W}_n$ for some bounded random vector $\varvec{W}_n$, when $n \rightarrow \infty$. The formalization of the above heuristic reasoning is given by the Kesten–Stigum limit theorem for super-critical multitype branching processes (see Kesten and Stigum 1966a, b, 1967). If $\varvec{W}_n=\varvec{Z}_n/\mu ^n$ then there exists a scalar random variable $W \ne 0$ such that, with probability one,

$$\begin{aligned} \lim _{n\rightarrow \infty } \varvec{W}_n = W \,\varvec{u} \end{aligned}$$

(4)

where $\varvec{u}$ is the right eigenvector corresponding to the malthusian parameter $\mu$ and

$$\begin{aligned} \mathbf {E}(W|\varvec{Z}_0)=\varvec{v}^{\mathrm {t}} \varvec{Z}_0 \end{aligned}$$

(5)

where $\varvec{v}$ is the left eigenvector corresponding to the malthusian parameter $\mu$. The vectors $\varvec{u}$ and $\varvec{v}$ may be normalized so that $\varvec{v}^{\mathrm {t}}\varvec{u}=1$ and $\varvec{1}^{\mathrm {t}}\varvec{u}=1$ where ${}^{\mathrm {t}}$ denotes the transpose of a vector. Moreover, under the assumption that $\varvec{M}$ is nonnegative [which is satisfied by the phenotypic model (18)], the right and left eigenvectors corresponding to the malthusian parameter are nonnegative.

The normalization of right eigenvector $\varvec{u}=(u_0,\ldots ,u_R)$ implies that $\sum _r u_r=1$, and therefore one has the “law of convergence of types” (see Kurtz et al. 1994)

$$\begin{aligned} \lim _{n\rightarrow \infty } \dfrac{\varvec{Z}_n}{|\varvec{Z}_n|} = \varvec{u} \,, \end{aligned}$$

(6)

where $|\varvec{Z}_n|=\sum _r Z_n^r$ is the total population at the nth generation and the equality holds almost surely. Equation (6) asserts that the asymptotic proportion of a replicative class r converges almost surely to the constant value $u_r$.

In particular, Eq. (6) implies that the malthusian parameter is the asymptotic relative growth rate of the population

$$\begin{aligned} \mu = \lim _{n\rightarrow \infty } \dfrac{|\varvec{Z}_{n}|}{|\varvec{Z}_{n-1}|} = \lim _{n\rightarrow \infty } \dfrac{1}{|\varvec{Z}_{n-1}|} \,\sum _{j=1}^{|\varvec{Z}_{n-1}|} \#[\,j\,] \end{aligned}$$

(7)

since $|\varvec{Z}_{n-1}|$ may be interpreted as the set of “parental particles” of the particles in the nth generation and $|\varvec{Z}_{n}|$ is the sum of the “progeny sizes” $\#[\,j\,]$ of the “parental particles” j from the previous generation.

Now consider the quantitative random variable $\rho$ defined on the set of classes $\{0,\ldots ,R\}$ and having probability distribution $(u_0,\ldots ,u_R)$, called the asymptotic distribution of classes. When the classes are indexed by their expectation values, the variable $\rho$ associates to a random particle its expected class

$$\begin{aligned} \mathbf {P}(\rho =r)=u_r \,. \end{aligned}$$

Therefore, one can define the average reproduction rate of the population as

$$\begin{aligned} \langle \rho \rangle = \sum _{r=0}^R r \, u_r \,. \end{aligned}$$

(8)

Using Eqs. (4), (5), (6) one can show that the average reproduction rate is equal to the malthusian parameter:

$$\begin{aligned} \langle \rho \rangle = \mu \,. \end{aligned}$$

(9)

The average population size at the nth generation is $|\langle \varvec{Z}_n \rangle | = \sum _{r=0}^R \langle Z^r_n \rangle$. Then for $n\rightarrow \infty$, Eq. (4) gives $|\langle \varvec{Z}_n \rangle | \approx \mu ^n |\langle \varvec{W}_n \rangle | \approx \mu ^n \langle W \rangle$ and so

$$\begin{aligned} \mu = \lim _{n\rightarrow \infty }\dfrac{|\langle \varvec{Z}_{n} \rangle |}{|\langle \varvec{Z}_{n-1} \rangle |} \end{aligned}$$

(10)

On the other hand, from the definition of mean matrix and its form (18), one has

$$\begin{aligned} |\langle \varvec{Z}_n \rangle | = |\varvec{M}\,\langle \varvec{Z}_{n-1} \rangle | =\sum _{r=0}^R r\,\langle Z^r_{n-1} \rangle \,. \end{aligned}$$

Now dividing by $|\langle \varvec{Z}_{n-1} \rangle |$ and taking the limit $n\rightarrow \infty$ gives

$$\begin{aligned} \mu = \lim _{n\rightarrow \infty }\dfrac{|\langle \varvec{Z}_{n} \rangle |}{|\langle \varvec{Z}_{n-1} \rangle |} = \lim _{n\rightarrow \infty }\sum _{r=0}^R r \,\dfrac{\langle Z^r_{n-1}\rangle }{|\langle \varvec{Z}_{n-1} \rangle |} = \sum _{r=0}^R r \, u_r = \langle \rho \rangle \end{aligned}$$

where here we used Eqs. (5) and (6) in the third equality from left to right.

In analogy with the characterization of the malthusian parameter as given by Eq. (7), one may define the asymptotic populational variance

$$\begin{aligned} \sigma ^2 = \lim _{n\rightarrow \infty } \dfrac{1}{|\varvec{Z}_{n-1}|} \, \sum _{j=1}^{|\varvec{Z}_{n-1}|} \#[\,j\,]^2 - \mu ^2 \end{aligned}$$

(11)

and in analogy with the mean reproduction rate, one may define the (squared) phenotypic diversity as

$$\begin{aligned} \sigma _{\rho }^2 = \langle \rho ^2 \rangle - \langle \rho \rangle ^2 \end{aligned}$$

(12)

By decomposing the sum in Eq. (11) according to the classes r, one obtains

$$\begin{aligned} \sum _{j=1}^{|\varvec{Z}_{n-1}|} \#[\,j\,]^2 = \sum _{r=0}^R \sum _{j_r=1}^{Z_{n-1}^r} \#[\,j_r\,]^2 \end{aligned}$$

where $j_r$ runs over the particles of class r for $r=0,\ldots ,R$ and $\#[\,j_r\,]$ are independent random variables assuming nonnegative values with probability distribution $t_r$, called fitness distribution of class r.

Denoting the variance of the fitness distribution $t_r$ by $\sigma ^2_{r}$, one may write the limit in Eq. (11) as

$$\begin{aligned} \sigma ^2= & {} \lim _{n\rightarrow \infty } \dfrac{1}{|\varvec{Z}_{n-1}|} \, \sum _{j=1}^{|\varvec{Z}_{n-1}|} \#[\,j\,]^2-\mu ^2 \\= & {} \lim _{n\rightarrow \infty } \dfrac{1}{|\varvec{Z}_{n-1}|} \,\sum _{r=0}^R \left[ Z^r_{n-1} \left( \dfrac{1}{Z^r_{n-1}} \sum _{j_r=1}^{Z_{n-1}^r}\#[\,j\,]^2 - r^2 \right) +Z^r_{n-1}\right] -\mu ^2 \\= & {} \lim _{n\rightarrow \infty } \dfrac{1}{|\varvec{Z}_{n-1}|} \,\sum _{r=0}^R (\sigma ^2_{r}+r^2)Z^r_{n-1}-\mu ^2 \end{aligned}$$

Then Eqs. (6), (9) and (12) give

$$\begin{aligned} \sigma ^2 = \sum _{r=0}^R (\sigma ^2_{r}+r^2 )\,u_r-\mu ^2 = \sum _{r=0}^R \sigma ^2_{r}\,u_r+\sigma ^2_{\rho } \end{aligned}$$

(13)

The difference between the asymptotic populational variance and the (squared) phenotypic diversity, called normalized populational variance, is the weighted average of the variances of the fitness distributions

$$\begin{aligned} \phi = \sigma ^2 - \sigma ^2_{\rho } = \sum _{r=0}^R \sigma ^2_{r}\,u_r \,. \end{aligned}$$

(14)

In particular, when the family of fitness distributions is the deterministic family the populational variance is exactly the phenotypic diversity (that is $\phi =0$). This is an expected result since the Delta distributions $t_r(k)=\delta _{rk}$ have zero variance, and hence the only source of fluctuation of the population size is due to its stratification into replicative classes, which is expressed by the phenotypic diversity.

B Mathematical Basis of the Phenotypic Model

Based on the general aspects of the phenomenon of viral replication described before, it is compelling to model it in terms of a branching process. At each replicative cycle, every parental particle in the replicative class r produces a random number of progeny particles that is independently drawn from the corresponding fitness distribution.

A fitness distribution is a member of a location-scale family of discrete probability distributions $t_r$ parameterized by the replicative classes ($r=0,\ldots ,R$) assuming nonnegative integer values and normalized so that the expectation value of $t_r$, defined as $\sum _{k} k\,t_r(k)$, is exactly r and $t_0(k)=\delta _{k0}$. Here $\delta _{kr}=1$ if $k=r$ and $\delta _{kr}=0$ if $k \ne r$. Therefore, each particle in the viral population is characterized by the mean value of its fitness distribution, called mean replicative capability. Viral particles with replicative capability equal to zero (0) do not generate progeny; viral particles with replicative capability one (1) generate one particle on average; viral particles with replicative capability two (2) generate two particles on average, and so on. Typical examples of location-scale families of discrete probability distributions that can be used as fitness distributions are:

• (a) The family of Deterministic (Delta) distributions: $t_r(k)=\delta _{kr}$.

• (b) The family of Poisson distributions: $t_r(k)=\mathrm {e}^{-r}\tfrac{r^k}{k!}$.

Note that in the first example, the replicative capability is completely concentrated on the mean value r – that is, the particles have deterministic fitness. On the other hand, in the second example the fitness is truly stochastic.

During the replication, each progeny particle always undergoes one of the following effects:

Deleterious effect::

the mean replication capability of the respective progeny particle decreases by one. Note that when the particle has capability of replication equal to 0, it will not produce any progeny at all.

Beneficial effect::

the replication capability of the respective progeny particle increases by one. If the mean replication capability of the parental particle is already the maximum allowed, then the mean replication capability of the respective progeny particles will be the same as the replicative capability of the parental particle.

Neutral effect::

the mean replication capability of the respective progeny particle remains the same as the mean replication capability of the parental particle.

To define which effect will occur during a replication event, probabilities d, b and c are associated, respectively, with the occurrence of deleterious, beneficial and neutral effects. The only constraints these numbers should satisfy are $0\leqslant d,b,c\leqslant 1$ and $b+c+d=1$. In the case of in vitro experiments with homogeneous cell populations, the probabilities c, d and b essentially refer to the occurrence of mutations.

The probability generating function (PGF) of the phenotypic model with $b=0$ and $t_r(k)=\delta _{kr}$ is (see Antoneli et al. (2013a, b) for details):

$$\begin{aligned} f_0(z_0,z_1,\ldots ,z_R)&= 1 n\nonumber \\ f_1(z_0,z_1,\ldots ,z_R)&= dz_0+cz_1 \nonumber \\ f_2(z_0,z_1,\ldots ,z_R)&= (dz_1+cz_2)^2 \nonumber \\&\vdots \nonumber \\ f_R(z_0,z_1,\ldots ,z_R)&= (dz_{R-1}+cz_R)^R \end{aligned}$$

(15)

Note that the functions $f_r(z_0,z_1,\ldots ,z_R)$ are polynomials whose coefficients are exactly the probabilities of the binomial distribution $\mathrm {binom}(k;r,1-d)$. The PGF in the case with general beneficial effects and with a general family of fitness distribution (which reduces to the previous PGF when $b=0$ and $t_r(k)=\delta _{kr}$) is given by.

$$\begin{aligned} f_0(z_0,z_1,\ldots ,z_R)&= 1 \nonumber \\ f_1(z_0,z_1,\ldots ,z_R)&= \sum _{k=0}^\infty \,t_1(k)\, (dz_0+cz_1+bz_2)^k \nonumber \\ f_2(z_0,z_1,\ldots ,z_R)&= \sum _{k=0}^\infty \,t_2(k)\, (dz_1+cz_2+bz_3)^k \nonumber \\&\vdots \nonumber \\ f_R(z_0,z_1,\ldots ,z_R)&= \sum _{k=0}^\infty \,t_R(k)\, (dz_{R-1}+(c+b)z_R)^k \end{aligned}$$

(16)

Note that in the last equation, the beneficial effect acts like the neutral effect. This is a kind of “consistency condition” ensuring that the populational replicative capability is, on average, upper bounded by R. Even though it is possible that a parental particle in the replicative classes R eventually has more than R progeny particles when $t_r$ is not deterministic, the average progeny size is always R.

Finally, it is easy to see that the PGF of the two-dimensional case of the phenotypic model with $b=0$ and $z_0=1$ (and ignoring $f_0$) reduces to

$$\begin{aligned} f(z)~=~\sum _{k=0}^\infty \,t(k)\, ((1-c)+cz)^k~=~\sum _{k=0}^\infty \,t(k)\, (1-c(1-z))^k \,. \end{aligned}$$

(17)

This is formally identical to the PFG of the single-type model proposed by [Demetrius et al. 1985, p. 255, eq. (49)] for the evolution of polynucleotides. In their formulation, $c=p^\nu$ is the probability that a given copy of a polynucleotide is exact, where the polymer has chain length of $\nu$ nucleotides and p is the probability of copying a single nucleotide correctly. The replication distributiont(k) provides the number of copies a polynucleotide yields before it is degraded by hydrolysis.

A remarkable property of the phenotypic model that was fully explored in Antoneli et al. (2013a, b) is the fact that when $b=0$ the phenotypic model is “exactly solvable” in a very specific sense.

It is straightforward form the generating function (16), using formula (3), that the matrix of the phenotypic model is given by

$$\begin{aligned} \varvec{M}=\begin{pmatrix} 0 &{}\quad d &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0 \\ 0 &{}\quad c &{}\quad 2d &{}\quad 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0 \\ 0 &{}\quad b &{}\quad 2c &{}\quad 3d &{}\quad 0 &{}\quad \ldots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 2b &{}\quad 3c &{}\quad 4d &{}\quad \ldots &{}v 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 3b &{}\quad 4c &{}\quad \ldots &{}\quad 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{}\quad \vdots &{} \ddots &{}\quad Rd \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad (R-1)b &{}\quad R(c+b) \end{pmatrix} \,. \end{aligned}$$

(18)

Note that the mean matrix does depend on the fitness distributions $t_r$ only through their mean values, since $t_r$ are normalized to have the mean value r.

Assume for a moment that $b=0$ (hence $c=1-d$). Then the mean matrix becomes upper-triangular, and hence its eigenvalues are the diagonal entries $\lambda _r=r(1-d)$ and the malthusian parameter $\mu$ is the largest eigenvalue $\lambda _R$:

$$\begin{aligned} \mu =R(1-d) \,. \end{aligned}$$

(19)

Now suppose that $b \ne 0$ is small compared to d and c (hence $c=1-d-b$). Then spectral perturbation theory allows one to write the malthusian parameter $\mu$ as a power series

$$\begin{aligned} \mu = \mu _0 + \mu _1 b + \mu _2 b^2 + \cdots \end{aligned}$$

where $\mu _0$ is the malthusian parameter for the case $b=0$ and $\mu _j$ are functions of the form $R\,\tilde{m}_j(d,R)$. A lengthy calculation (see Antoneli et al. 2013b) gives the following result:

$$\begin{aligned} \mu = R \left( (1-d) + (R-1)\dfrac{d}{1-d}\,b + \varvec{O}(b^2)\right) \,. \end{aligned}$$

(20)

Let us return to the case $b=0$ and consider the eigenvectors corresponding to the malthusian parameter $\mu$. The right eigenvector $\varvec{u}=(u_0,\ldots ,u_R)$ and the left eigenvector $\varvec{v}=(v_0,\ldots ,v_R)$ may be normalized so that $\varvec{v}^{\mathrm {t}}\varvec{u}=1$ and $\varvec{1}^{\mathrm {t}}\varvec{u}=1$, where ${}^{\mathrm {t}}$ denotes the transpose of a vector. In Antoneli et al. (2013b), it is shown that the normalized right eigenvector $\varvec{u}=(u_0,\ldots ,u_R)$ is given by

$$\begin{aligned} u_r=\left( {\begin{array}{c}R\\ r\end{array}}\right) \, (1-d)^r \, d^{R-r} \,. \end{aligned}$$

(21)

The fact that $\varvec{u}$ is a binomial distribution is not accidental. Indeed, it can be shown that $\varvec{u}$ is the probability distribution of a quantitative random variable $\rho$ defined on the set of replicative classes $\{0,\ldots ,R\}$, called the asymptotic distribution of classes, such that $u_r=\mathrm {binom}(r;R,1-d)$ gives the limiting proportion of particles in the rth replicative class. Finally, when $b \ne 0$ is small, spectral perturbation theory ensures that

$$\begin{aligned} u_r=\left( {\begin{array}{c}R\\ r\end{array}}\right) \, (1-d)^r \, d^{R-r} + \varvec{O}(b) \,. \end{aligned}$$

(22)

The phenotypic model is completely specified by the choice of the two probabilities b and d (since $c=1-b-d$), the maximum replicative capability R and a choice of a location-scale family of fitness distributions. Independent of the choice of family of fitness distributions, the parameter space of the model is the set $\triangle ^2 \times \{R\in \mathbb {N}:R \geqslant 1\}$, where $\triangle ^2=\{(b,d)\in [0,1]^2: b+d \leqslant 1\}$ is the two-dimensional simplex (see Fig. 10).

Fig. 10

Parameter space of the phenotypic model. The blue line is boundary $b+d=1$. The red, green and magenta curves are the critical curves $\mu (b,d,R)=1$ for $R=2,3,4$, respectively

In this parameter space, one can consider the critical curves$\mu (b,d,R)=1$, where $\mu (b,d,R)$ is the malthusian parameter as a function of the parameters of the phenotypic model. For each fixed R, the corresponding critical curve is independent of the fitness distributions and represents the parameter values (b, d) such that the branching process is critical. Moreover, each curve splits the simplex into two regions representing the parameter values where the branching process is super-critical (above the curve) and sub-critical (below the curve).

One of the main results of Antoneli et al. (2013b) is a proof of the lethal mutagenesis criterion (Bull et al. 2007) for the phenotypic model, provided one assumes that all fitness effects are of a purely mutational nature. Recall that (Bull et al. 2007) assumes that all mutations are either neutral or deleterious and consider the mutation rate$U=U_d+U_c$, where the component $U_c$ comprises the purely neutral mutations and the component $U_d$ comprises the mutations with a deleterious fitness effect. Furthermore, $R_{\mathrm {max}}$ denotes the maximum replicative capability among all particles in the viral population. The lethal mutagenesis criterion proposed by Bull et al. (2007) states that a sufficient condition for extinction is

$$\begin{aligned} R_{\mathrm {max}}\,\mathrm {e}^{-U_d} < 1 \,. \end{aligned}$$

(23)

According to (Bull et al. 2007, 2008), $\mathrm {e}^{-U_d}$ is both the mean fitness level and also the fraction of offspring with no non-neutral mutations. Moreover, in the absence of beneficial mutations and epistasis (Kimura and Maruyama 1966) the only type of non-neutral mutation are the deleterious mutations. Therefore, in terms of fitness effects, the probability $\mathrm {e}^{-U_d}$ corresponds to $1-d=c$. Since the evolution of the mean matrix depends only on the expected values of the fitness distribution $t_r$, it follows that $R_{\mathrm {max}}$ corresponds to R. That is, the lethal mutagenesis criterion of (23) is formally equivalent to extinction criterion

$$\begin{aligned} R(1-d) < 1 \end{aligned}$$

(24)

which is exactly the condition for the phenotypic model to become sub-critical. Formula (20) for the malthusian parameter provides a generalization of the extinction criterion (24) without the assumption that that all effects are either neutral or deleterious. If $b>0$ is sufficiently small (up to order $\varvec{O}(b^2)$) and

$$\begin{aligned} R \left( (1-d) + (R-1)\dfrac{bd}{1-d}\right) < 1 \end{aligned}$$

(25)

then, with probability one, the population becomes extinct in finite time.

On the other hand, a deeper exploration of the implications of nonzero beneficial effects allowed for the discovery of a non-extinction criterion. If $b>0$ is sufficiently small (up to order $\varvec{O}(b^2)$), R is sufficiently large ($R \geqslant 10$ is enough) and

$$\begin{aligned} R^3 \, b > 1 \end{aligned}$$

(26)

then, asymptotically almost surely, the population cannot become extinct by increasing the deleterious probability d toward its maximum value $1-b$ (see Antoneli et al. 2013b for details). In other words, a small increase in the beneficial probability may have a drastic effect on the extinction probabilities, possibly rendering the population impervious to become extinct by lethal mutagenesis (i.e., by the increase in deleterious effects).

In the theory of multitype branching processes, there are several variations as follows: continuous time, age dependent, self-regulated, etc. (see Athreya and Ney 1972; Harris 1963; Kimmel and Axelrod 2002). The implementation of a variation of the theory of multitype branching process accounting for the notions of evolutionary entropy and directionality theory (see Dietz 2005; Demetrius 2013) could be useful for studies on viral evolution. In this case, the malthusian parameter $\mu$, which is the dominant eigenvalue of the mean matrix, could be expressed as the sum of two terms

$$\begin{aligned} \mu = H + \varPhi \,. \end{aligned}$$

The quantity H is called evolutionary entropy and $\varPhi$ is called the reproductive potential (Demetrius 2013). An interesting direction to follow would be to develop an extinction criterion based on evolutionary entropy instead of the malthusian parameter.

C The Deterministic Selection Equation

According to (Demetrius et al. 1985; Demetrius 1985, 1987), one may associate to a multitype branching process a system of difference (or ordinary differential) equations, called selection equations, on the space of discrete probability distributions $\triangle ^{R+1}=\{\varvec{p}\in \mathbb {R}^{R+1}:p_j\geqslant 0;\sum _{j}p_j=1\}$ over the finite state set $\{0,\ldots ,R\}$. Given a discrete multitype branching process $\varvec{Z}_n$, then the expectation values $\langle \varvec{Z}_n\rangle$ satisfy $\langle \varvec{Z}_n\rangle =\varvec{M}^n\varvec{Z}_0$, with $\varvec{M}$ being the mean matrix of $\varvec{Z}_n$. Hence $\varvec{Z}_n$ is given by iteration of the difference equation $\varvec{z}_n= \varvec{M}\varvec{z}_{n-1}$. This yields a discrete-time selection equation by normalizing the difference equation, thereby obtaining

$$\begin{aligned} \varvec{x}_n = \dfrac{1}{\varvec{1}^\mathrm {t}\varvec{M}\varvec{x}_{n-1}}\varvec{M}\varvec{x}_{n-1} \end{aligned}$$

(27)

where $\varvec{1}=(1,\ldots ,1)$. Then, passing (27) to continuous time one obtains a continuous-time selection equation

$$\begin{aligned} \dot{\varvec{x}} = [\varvec{M}\varvec{x}-\varvec{x}(\varvec{1}^\mathrm {t}\varvec{M}\varvec{x})] \dfrac{1}{\varvec{1}^\mathrm {t}\varvec{M}\varvec{x}} \,. \end{aligned}$$

(28)

Multiplying the right hand side of Eq. (28) with the factor $\varvec{1}^\mathrm {t}\varvec{M}\varvec{x}$, which is always strictly positive on $\triangle ^{R+1}$, corresponds to a change in velocity (re-scaling time) and so, the solutions of (28) are the same as the solutions of

$$\begin{aligned} \dot{\varvec{x}} = \varvec{M}\varvec{x}-\varvec{x}(\varvec{1}^\mathrm {t}\varvec{M}\varvec{x}) \end{aligned}$$

(29)

It follows from general considerations (see Demetrius et al. 1985; Demetrius 1985, 1987) that Eq. (29) has a unique global stable equilibrium on $\triangle ^{R+1}$ given by the normalized right eigenvector $\varvec{u}$ of $\varvec{M}$ corresponding to its largest eigenvalue $\mu$. In this sense, the deterministic selection equation yields a description of the evolution of the normalized mean values of the corresponding stochastic model, thus defining a mean field (macroscopis) dynamics representing the infinite population limit of the branching process.

D The Power Law Distribution Family

It is typical to parameterize power law distributions by the exponents, which measures the “weight of the tail” of the distribution. However, we need to have a location-scale parameterized family in order to impose the same normalization as we have done for the other types of distributions. Therefore, we define the power law distribution with mean valuer by

$$\begin{aligned} \mathfrak {z}_r(k) = \frac{(k-1)^{s(r)}}{\zeta (s(r))} \end{aligned}$$

for $k=0,1,\ldots ,\infty$ and $r \geqslant 1$, where $\zeta (s)$ is the Riemann zeta function, defined for $s>1$, by

$$\begin{aligned} \zeta (s) = \sum _{n=1}^{\infty } \frac{1}{n^s} \end{aligned}$$

and the function s(r) is given by the inverse function of

$$\begin{aligned} r = \varphi (s) = \frac{\zeta (s-1)}{\zeta (s)}-1 \,. \end{aligned}$$

Namely, $s=\varphi ^{-1}(r)$ for $r\geqslant 1$ and hence when $1 \leqslant r < \infty$ the exponent s satisfies $3<s<2$. Moreover, the Laurent series expansion for $r\rightarrow \infty$ ($s \rightarrow 2$) is given by:

$$\begin{aligned} s(r) \approx 2 + \frac{6}{\pi ^2(1+r-C)} \,. \end{aligned}$$

(30)

The constant C in the previous formula is given by $C = [6 \gamma \pi ^2 - 36\,\zeta '(2)]/\pi ^4 \approx 0.6974$, where $\gamma$ is Euler’s constant and $\zeta '(2)$ is the derivative of $\zeta (s)$ evaluated at 2. Observe that when the mean value $r \geqslant 1$, the exponent $s<3$, and so the variance of $\mathfrak {z}_r(k)$ is infinite.

The implementation of the pseudo-random generation of samples from the distribution $\mathfrak {z}_r(k)$ in the ENVELOPE program is based on the algorithm of Devroye (1986) for the Zipf distribution on the positive integers, using formula 30 for the computation of the exponent s given the mean value r. Pseudo-random generation for the remaining fitness distributions was implemented using the standard library of C++ programing language (this library requires C++ (2011) or superior).

E Main Routines of the ENVELOPE Program