III Related Work: Evolutionary Theory and Modeling Here I provide background work in mathematical epidemiology and evolutionary theory. This provides the theoretical and contextual foundation for my genomic interests, and is important to highlight because there have been criticisms of large-scale genomic approaches for being short on insight, hypotheses, or elegance. My viral genomics interests, alternatively, follows naturally from mathematical and theoretical pursuits of mine. Theoretical Underpinnings. My interests in viral genomics are informed by theoretical work that predicts tradeoffs between traits involved in replication efficiency vs. outside-of-host survival. We have prepared a theoretical model that suggests that the optimal strategy for a virus in a given system will be the one where the ratio of efficiency (replication) traits to opportunity (survival) traits is maximized. In the case of HCV, this translates to a purported tradeoff between transmission characteristics and replication characteristics. Insofar as virulence is often tied to within-host replication, we might observe that selection on traits for improved transmission might come at the expense of disease phenotypes. To demonstrate how theory has informed these approaches, we’ll use a basic model of virus dynamics that describes the interaction between virus and hosts as originally described by Nowak1. The model describes a system with a set of basic assumptions on the virus-host interaction: no multiple infections. The dynamics can be described by a set of differential equations: !"!"= 𝜋 − 𝛽𝑆 𝑡 𝑉 𝑡 − 𝜇!𝑆(𝑡) [5]

!"!"= 𝛽𝑆 𝑡 𝑉 𝑡 − 𝛿𝐼(𝑡) [6]

!"!"= 𝜅𝐼 𝑡 − 𝜀𝑉(𝑡) [7]

Where is the death rate of uninfected hosts; is the death rate of infected hosts; is the replication rate of viruses; is the death rate of free virions. From the set of differential equations we can derive the expression for the basic reproduction number, , using a method original developed by ven den Driessche and Watmough(2006)2 and more recently by Blower (2007)3. Using the set of ordinary differential equations that define the model, we created by two 2 * 1 vectors, F and V, that represent new and transported infections:

𝐹 = 0 !"!!

0 0and 𝑉 = 𝛿 0

−𝜅 𝜀 [8], [9]

𝐹𝑉!! = 0 !"!!

0 0

!!

0!!"

!!

[10]

µh δ κε

R0

 

 

Then

𝑅! =!"#!!!"

[11] From this we can analyze virus dynamics as a product of the various host and virus-defined parameters. For these purposes, we’ll assume that only κ and ε are virally-determined. To calculate an evolutionary stable strategy, we can start with the reproductive rate: 𝑅! =

!"#!!!"

[12] Only the virus replication rate, κ, is a function of the virus survival rate, epsilon. So we have:

𝑅! 𝛿 = !!!!

!"!"!!!(!)

!! [13]

Therefore !!!!" !!!∗

= 0 if    !"!" !!!∗

=   !(!∗)!

[14] This translates to an optimum strategy for virus replication being the slope of a line that relates κ and ε (Figure 1) In this virus host-system, 𝜋 defines those host-dependent factors that provide opportunities for efficient creation of offspring. The virally determined efficiency traits are captured by the κ term and the virally-determined opportunity traits by the ε term.

Figure 5. Shifting optima. The optimal ratio of efficiency and opportunity values for given population of viruses moving from one environment to another is governed by changes in the maximum efficiency in a new environment and number of opportunities for efficient replication in a new host. 1A demonstrates a standard curve relating efficiency to (inverse) opportunity). In 1B represents the way the curve shifts in a new environment. For the purposes of this project, we can equate [k] to traits influencing replication and [ε] to traits influencing transmission (from Ogbunugafor et al. 2013, in preparation).

 

 

We believe that the use of theoretical models to drive our hypotheses represents an improvement over many large-scale genomic approaches in infectious diseases, where the goals are mostly data collection, followed by hypothesis generation. While hypothesis generation is a reasonable component of any large-scale genomics project, that we have buffered our aims with evolutionary theory indicates that the study is rooted in sound biology, with a theoretical basis upon which to test the findings in HCV. We can observe whether selection on certain traits (those affecting within-host replication or transmission) comes at the expense of other traits, including those that contribute to virulence. Epidemiological Modeling. In addition, I am also developing a pure epidemiological model for HCV transmission. Figure 6 shows the box model designating the flow of individuals and viruses in a realistic ecological setting (injection drug use). The model is unique among epidemiological models in how it models infected needles as a separate entity directly affected by biological properties of the viruses being transmitted. Not only will this model create testable predictions for how HCV prevalence would be affected by evolution, it can be retrofit to the data produced from the genomics study. Several of the tools used in Aim 2 can directly calculate parameters like the basic reproductive number, Ro, which can easily be fed into the epidemiological model.

Analytic equations dSdt

= π −γβS(t)I(t) Ni (t)Ni (t)+ Nu(t)"

#$

%

&'−µnS(t)

dIEdt

= γβS(t) Ni (t)Ni (t)+ Nu(t)"

#$

%

&'− IE (t)ω − IE (t)τ − IE (t)µn − IE (t)φ

dIL = IE (t)ω − IL (t)µL − IL (t)τ − IL (t)φ +T (t)ρdTdt

= τ IE (t)+τ IL (t)− ρT (t)−T (t) 1− ρ( )−µTT (t)

dRdt

= IE (t)φ +T (t) 1− ρ[ ]−µnR(t)

dNu

dt= π +εNi (t)−κNu(t)

dNi

dt= γζ IE (t)+ IL (t)[ ] Nu(t)

Ni (t)+ Nu(t)[ ]−κNi (t)−εNi (t)

Where S is the number of susceptible patients in a population, IE the number in an early infection indown, IL the number who have an established latent infection, T the number who have been treated, R the number of who have recovered, Nu the number of uninfected needles and Ni the number of infected needles.  

!!R!

!!!S! !!IE!!! !!!IL!!!

!!!Nu! !!Ni!

!!!T!!!

µn

IE (t)ω τ IL (t)

εNi (t)

γζNu(t)

Ni (t)+ Nu(t)[ ]

π B

κNu(t) κNi (t) φIE (t)φIL (t)

τ IE (t)

γβS(t) Ni (t)Ni (t)+ Nu(t)!

"#

$

%&

µn

T (t)[1− ρ]

µn

π SµLµn

T (t)ρ

Figure 2. Box model outlining HCV transmission dynamics (from Ogbunugafor et al., in preparation).  

 

 

Works Cited Van den Driessche, P., and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48.

Kajita, E., Okano, J.T., Bodine, E.N., Layne, S.P., and Blower, S. (2007). Modelling an outbreak of an emerging pathogen. Nat. Rev. Microbiol. 5, 700–709.

Nowak, M., and May, R.M. (2000). Virus Dynamics: Mathematical Principles of Immunology and Virology (Oxford University Press).