Emerging pathogens the epidemiology and evolution of species jumps by Woolhouse, Mark E. J.

Summary

We conclude that although pathogen emergence is inherently unpredictable, emerging pathogens tend to share some common traits, and that directly transmitted RNA viruses might be the pathogens that are most likely to jump between host species

Main Points

Understanding the epidemiology and evolutionary biology is important to understand emerging infectious diseases in humans, domestic animals, wildlife, and plant populations.

Understanding $R_0$

When $R_0 \lt 1$ then the majority of the new infections will likely be acquired from the original host population (or reservoir species) rather than when Error in rendering LaTeX when the infections are likely from within the new host. This is important when understanding the dynamics of an outbreak or epidemic, especially during spillovers. Additionally, when Error in rendering LaTeX or near 1 then there is a positive feedback loop allowing for evolutionary changes to occur for new host adaptation.

Drivers for change in $R_0$

• Changes in host ecology and environment
• Changes in host behaviour and movement
• Changes in host phenotype
  • Example here is immunosuppression in cancer or HIV patients and opportunistic infections
  • This is likely some of the cause of the rise in fungal infections
• Changes in host genetics
• Changes in pathogen genetics

Epidemic Thresholds and $R_0$

Below is the probability of a major epidemic as related to values of $R_0$ and $I_0$

$$P_{epidemic} = 1 = \Big(\frac{1}{R_0}\Big)^{I_0}$$

Now we can consider the probability of adaptation

$$P_{adaptation} = \frac{\mu R_0}{1 - R_0}$$

Where $\mu$ is the probability of that the required genetic change occurs during a single infection.

Of course the final epidemic size is the solution of:

$$I_{final} = N - (N-I_0) exp\Big[ \frac{-R_0I_{final}}{N}\Big]$$

Which can be solved via:

$$1 + \frac{1}{R_0 W[-R_0exp(-R_0)]}$$

epi_finalsize <- function(r0) {
	1+1/r0*lambertW(-r0*exp(-r0))
}

# Where:
#' @details Compute the Lambert W function of z.  This function satisfies
#' W(z)*exp(W(z)) = z, and can thus be used to express solutions
#' of transcendental equations involving exponentials or logarithms.
#' The Lambert W function is also available in
#' Mathematica (as the ProductLog function), and in Maple and Wolfram.
#'
#' @references Corless, Gonnet, Hare, Jeffrey, and Knuth (1996), "On the Lambert
#' W Function", Advances in Computational Mathematics 5(4):329-359
#' @author Nici Schraudolph <schraudo at inf.ethz.ch> (original
#'   version (c) 1998), Ben Bolker (R translation)
#'   See <https://stat.ethz.ch/pipermail/r-help/2003-November/042793.html>
lambertW = function(z, b=0, maxiter=10, eps=.Machine$double.eps,
										min.imag=1e-9) {

	if (any(round(Re(b)) != b))
		stop("branch number for W must be an integer")
	if (!is.complex(z) && any(z<0)) z=as.complex(z)
	## series expansion about -1/e
	##
	## p = (1 - 2*abs(b)).*sqrt(2*e*z + 2);
	## w = (11/72)*p;
	## w = (w - 1/3).*p;
	## w = (w + 1).*p - 1
	##
	## first-order version suffices:
	##
	w = (1 - 2*abs(b))*sqrt(2*exp(1)*z + 2) - 1
	## asymptotic expansion at 0 and Inf
	##
	v = log(z + as.numeric(z==0 & b==0)) + 2*pi*b*1i;
	v = v - log(v + as.numeric(v==0))
	## choose strategy for initial guess
	##
	c = abs(z + exp(-1));
	c = (c > 1.45 - 1.1*abs(b));
	c = c | (b*Im(z) > 0) | (!Im(z) & (b == 1))
	w = (1 - c)*w + c*v
	## Halley iteration
	##
	for (n in 1:maxiter) {
		p = exp(w)
		t = w*p - z
		f = (w != -1)
		t = f*t/(p*(w + f) - 0.5*(w + 2.0)*t/(w + f))
		w = w - t
		if (abs(Re(t)) < (2.48*eps)*(1.0 + abs(Re(w)))
				&& abs(Im(t)) < (2.48*eps)*(1.0 + abs(Im(w))))
			break
	}
	if (n==maxiter) warning(paste("iteration limit (",maxiter,
																") reached, result of W may be inaccurate",sep=""))
	if (all(Im(w)<min.imag)) w = as.numeric(w)
	return(w)
}

Compatibility

Pathogens and new hosts need to be compatible (i.e., target receptors have to be present in the case of viruses, the right resources need to be available for fungi or bacteria). Pathogens often have highly selective host ranges. If receptors are conserved across species then they make potentially good targets and explain why certain pathogens have broader host ranges.

Pathogens must overcome the “species barrier” which means that when jumping between species the minimum dose required for infection likely increases (e.g., it takes more virus particles to infect a human with a rodent virus than it does for rodent to rodent transmission).

The pathogen must also be transmissible between members of the same species.

Evolution and host adaptation

Because of a lack of shared history, emerging pathogens do not have evolved limits on pathogenicity and host susceptibility.

Adaptation to a new host can occur rapidly to change vectors.

The probability of successful adaption ocurring depends on several factors:

• the number of primary infections
• The initial $R_0$ of the infection in the new host
• The number of mutations or other genetic changes (substitutions) required
• The likelihood of these changes occurring and how $R_0$ (or more generally the fitness) changes at each substitution change

The probability of each rare evolutionary step is proportional to the expected size of the initial outbreak, thus the number of opportunities to occur. Probability emergence increases linearly with $I_0$ but is more sensitivity to $R_0$

Combating emerging pathogens

Questions

• Can we put this in a spatiotemporal framework? Thinking of pulsed immigration and spatial/temporal variation with these dynamics for emergence.
• Similarly, in a metapopulation model rather than the initial spillover event?

Code/ Supplemental Data

Implications for Infectious Diseases