Pathogen Evolution When Transmission and Virulence are Stochastic

Evolutionary processes are inherently stochastic, since we can never know with certainty exactly how many descendants an individual will leave, or what the phenotypes of those descendants will be. Despite this, models of pathogen evolution have nearly all been deterministic, treating values such as transmission and virulence as parameters that can be known ahead of time. We present a broadly applicable analytic approach for modeling pathogen evolution in which vital parameters such as transmission and virulence are treated as random variables, rather than as fixed values. Starting from a general stochastic model of evolution, we derive specific equations for the evolution of transmission and virulence, and then apply these to a particular special case; the SIR model of pathogen dynamics. We show that adding stochasticity introduces new directional components to pathogen evolution. In particular, two kinds of covariation between traits emerge as important: covariance across the population (what is usually measured), and covariance between random variables within an individual. We show that these different kinds of trait covariation can be of opposite sign and contribute to evolution in very different ways. In particular, probability covariation between random variables within an individual is sometimes a better way to capture evolutionarily important tradeoffs than is covariation across a population. We further show that stochasticity can influence pathogen evolution through directional stochastic effects, which results from the inevitable covariance between individual fitness and mean population fitness.

Authors and Affiliations

1. Department of Biological Sciences, Texas Tech University, 2901 Main St., Lubbock, TX, 79409, USA

   Pooya Aavani & Sean H. Rice

Corresponding author

Correspondence to Pooya Aavani .

Editors and Affiliations

1. Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco

   Khalid Hattaf

2. Department of Mathematics, Gauhati University, Guwahati, India

   Hemen Dutta

Appendix 1

We will use the following identities in manipulating frequency and probability operations:

(51)

The following theorem is useful for manipulating covariance of products:

Theorem 1

1. (i)

   For three arbitrary random variables a, b, and c we have the following identity

   

   (52)
2. (ii)

   If c is independent from a, b, and ab, Eq. (52) collapses to:

   

   (53)

Proof

1. (i)

   We extract the right side of the Eq. (52)

   
2. (ii)

   If c is independent of a, b, and ab then , and we have

   

Note that the results of (1) as well can be applied to frequency operation substituting for and \(\overline{*}\) for \(\widehat{*}\).

Recall that for the SIR model the absolute finesses of pathogen strains are independent from each other. With this assumption, the covariance between fitness of two strains will be zero, i.e, and .

The third term of the Eq. (2) is expanded as a sum of two other terms in Eq. (38). Because of the stochastic independency, the covariance between the fitness of two different strains becomes zero and therefore the second term of the Eq. (38) becomes zero and we obtain:

(54)

The fourth term of the Eq. (2) is expanded as a sum two other terms in Eq. (39). By part (ii) of Theorem (1), the second term of the Eq. (39) become zero and the first term simplified as follows:

(55)

Therefore, using the results of Eqs. (54) and (55) we simplify Eq. (2) as follows:

(56)

Now we try to obtain each term of the Eq. (6). Here, we use Eq. (3) and similar equations will be obtained using Eq. (4).

Using Eq. (5), we expand the first term on the right-hand side of the Eq. (3) as:

(57)

The second term on the right-hand side of the Eq. (3) can be expanded as:

(58)

The third term of the Eq. (3) is expanded as:

(59)

The fourth term on the right-hand side of the Eq. (3) is expanded below:

(60)

The fifth term on the right-hand side of the Eq. (3) can be expanded as follows:

(61)

The sixth term on the right-hand side of the Eq. (3) can be expanded as follows:

(62)

We expand the seventh term on the right-hand side of the Eq. (3) in two steps:

(63)

(64)

Equation (63) can be further expanded as follows:

(65)

We expand Eq. (64) as follows:

(66)

(67)

Finally, the eights term on the right-hand side of the Eq. (3) is expanded below:

(68)

Appendix 2

From model (22), we derive the equation for fitness of pathogen strain i as:

Note that in SIR model the pathogen strains are stochastically independent in fitness. By substituting Eq. (69) in Eqs. (3) and (4) we derive Eq. (24). We present Eq. (24) with the tensor notations. For the strain i, we define the random variables \(\phi ^o_{1}=\beta \), \(\phi ^o_{2}=\nu \), \(\phi ^o_{3}=d\), \(\phi ^o_{4}=c\). The covariance terms on the right-hand sides of Eqs. (24) can be grouped into \(4 \times 4\) degree 2 and \(4 \times 4 \times 4\) degree 3 tensors (note that a tensor of degree 2 is just a matrix). The elements of each tensor can be shown as follows:

where \(i=1...4, \quad j=1...4, \quad k=1...4\). Also we obtain:

The tensor inner products on the right-hand side of Eq. (24) is 4 by 1 vector and can be written as follows:

Now we try to derive the Eq. (56) for each phenotype \(\beta \), \(\nu \), d, and c where w is function of those variables and obtained from Eq. (69). Here we only show the detailed derivation of \(\widehat{\Delta \overline{\nu }}\) and others can be obtained similarly.

Using Eq. (69), we expand the first term on the right-hand side of the Eq. (56) as:

(78)

The second term on the right-hand side of the Eq. (56) can be expanded as:

(79)

The third term of the Eq. (56) is expanded as:

(80)

(81)

The fourth term on the right-hand side of the Eq. (56) is derived below:

(82)

(83)

Also, by substituting \(\nu \) in \(\phi \) and using Eq. (69), the fifth and sixth terms on the right-hand side of the Eq. (56) are expanded as follows:

(84)

(85)

(86)

(87)

The following equation shows the change in the average mean phenotype of the virulence over one generation. To track where each term comes from, we separate the corresponding terms by writing the equation numbers over them:

(88)

Note that c and d enter into Eq. (69) in the same way as does \(\nu \). We can thus derive equation \(\widehat{\Delta \overline{c}}\) and \(\widehat{\Delta \overline{d}}\) by substituting. Because \(\beta \) enters Eq. (69) as \(S\beta \), the equation for \(\widehat{\Delta \overline{\beta }}\) is structured a bit differently. We can use the same approach to derive the equations for the change in the average mean phenotype of the transmission over one generation:

(89)

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