Reading notes

Summary

SSematimba et al examine the per contact probability rates between poultry farms in the Netherlands in 2003 outbreaks of H7N7. H7N7 is a highly pathogenic strain for poultry. They used a series of probabilistic arguments backed by contact tracing data and some phylogenetic analysis to examine transmission routes. They found a little unsurprisingly that egg transport represents a very high probability of introducing infection to a new farm.

Main Points

![[Pasted image 20241025132811.png]]

Abstract

Estimates of the per-contact probability of transmission between farms of Highly Pathogenic Avian Influenza virus of H7N7 subtype during the 2003 epidemic in the Netherlands are important for the design of better control and biosecurity strategies. We used standardized data collected during the epidemic and a model to extract data for untraced contacts based on the daily number of infectious farms within a given distance of a susceptible farm. With these data, we used a maximum likelihood estimation approach to estimate the transmission probabilities by the individual contact types, both traced and untraced. The estimated conditional probabilities, conditional on the contact originating from an infectious farm, of virus transmission were: 0.000057 per infectious farm within 1 km per day, 0.000413 per infectious farm between 1 and 3 km per day, 0.0000895 per infectious farm between 3 and 10 km per day, 0.0011 per crisis organisation contact, 0.0414 per feed delivery contact, 0.308 per egg transport contact, 0.133 per other-professional contact and, 0.246 per rendering contact. We validate these outcomes against literature data on virus genetic sequences for outbreak farms. These estimates can be used to inform further studies on the role that improved biosecurity between contacts and/or contact frequency reduction can play in eliminating between-farm spread of the virus during future epidemics. The findings also highlight the need to; 1) understand the routes underlying the infections without traced contacts and, 2) to review whether the contact-tracing protocol is exhaustive in relation to all the farm’s day-to-day activities and practices.

Notes

They set out the following sets of equations and estimated using their contact rate data: $$\text{Prob of farm infected} = \big ( 1 - \Pi_{all i}(1-p_i)^{{\Sigma_dw_dC_{i,d}}{Infected}}\big)$$ And then the probability of a farm escaping infection is then $$\big ( (1 - p_i)^{{\Sigma (1 - w_d)C_{i,d}}{Inf}}\big)$$ And then the probability of a farm escaping infection during an outbreak is:

$$\big( \Pi_{all \text{i}}(1-p+i)^{C_i{Escape}}\big)$$

“The estimated conditional probabilities, conditional on the contact originating from an infectious farm, of virus transmission were: 0.000057 per infectious farm within 1 km per day, 0.000413 per infectious farm between 1 and 3 km per day, 0.0000895 per infectious farm between 3 and 10 km per day, 0.0011 per crisis organisation contact, 0.0414 per feed delivery contact, 0.308 per egg transport contact, 0.133 per other-professional contact and, 0.246 per rendering contact. We validate these outcomes against literature data on virus genetic sequences for outbreak farms.” (Ssematimba et al., 2012, p. 1)

“Following the detection of the first outbreak, a control programme, as stipulated by the European Union, was implemented. This programme consisted of stamping out of infected flocks, movement restrictions and establishment of protection and surveillance zones.” (Ssematimba et al., 2012, p. 1)

“However, in comparison with preventive culling, emergency vaccination would have the important disadvantage that its effect suffers from a 7 to 14 days protection delay 12]. This delay would prolong the time until epidemic control is obtained especially in the high density poultry areas (de Jong and Hagenaars [13] and the references therein).” ([Ssematimba et al., 2012, p. 1)

“Plausible mechanisms include movements of humans (professional and non-professional visitors, employees and farmers themselves), vehicular traffic (for example, delivery trucks), other fomites (such as tools, cell phones and shared farm equipment) and other vectors such as wind, rodents and insects 9,14–17]. These transmission events involve transportation of the virus either in contaminated litter, faeces or skin and feathers that can colloid on the fomites or the vectors’ body. Therefore, in order to better control neighbourhood transmission, we need to understand deeper the steps involved in the whole virus dissemination process; a quite complex task.” ([Ssematimba et al., 2012, p. 2)

“the probability of HPAI virus transmission may be contact-specific but will also depend on the contact patterns” (Ssematimba et al., 2012, p. 2)

“They found an increased risk of HPAI virus introduction in layer-finisher type poultry compared to other poultry types. Their analysis gave some clues on the risk factors for HPAI virus introduction such as poultry type and flock size.” (Ssematimba et al., 2012, p. 2)

“Our analysis aims to give quantitative insight into the role of the different between-farm contacts in the spread of the virus during an epidemic.” (Ssematimba et al., 2012, p. 2)

“This dataset captured information on a total of 614 visits originating from 203 infectious farms. Out of these visits, 381 were to infected farms. The total number of receiving farms was 325 of which 149 were ultimately infected.” (Ssematimba et al., 2012, p. 2)

“From this dataset we selected visits to a farm that occurred up to seven days prior to and excluding its day of suspicion. For these contacts, we only considered same-day visits i.e., those that occurred on the same day that the person had visited an infectious farm.” (Ssematimba et al., 2012, p. 2)

“A farm was deemed exposed if the visit occurred during the period when the virus was likely to have been introduced onto the receiving farm, here referred to as the potential virus-introduction period.” (Ssematimba et al., 2012, p. 2)

“We quantified the contribution of the different contacts to the epidemic in terms of the number of new infections that they may have caused. This was obtained by multiplying their estimated per-contact probability with their frequency.” (Ssematimba et al., 2012, p. 3)

“For the base model, we used a uniform distribution to obtain wd ~ 1 7. In other words, we assumed that each of the seven days of the probable period of virus introduction was equally likely to be the actual day of virus introduction.” (Ssematimba et al., 2012, p. 3)

“We assessed two other distributions in which the estimated weighting factors wd were adjusted to sum to one over the 7-day period, namely; 1) a distribution in which the probability is decreasing exponentially over the 7-day period at a rate determined by the survival of HPAI virus in manure (in this case 14 days 23]) and, 2) a unimodal distribution with the most likely day being 4 days prior to the day of clinical suspicion.” ([Ssematimba et al., 2012, p. 3)

“In this way, we used the genetic data to validate the estimated probabilities per contact: too few or too many genetic matches would cast doubt on the estimated probabilities.” (Ssematimba et al., 2012, p. 4)

“For those pairs (i.e., with complete genetic information), we compared their genetic sequences to ascertain which ones were sufficiently ‘‘matching’’ for transmission between A and B not to be unlikely. The number of genetically matching pairs, minus an estimate of the expected number of ‘‘bychance’’ genetic matches, was then compared to the predicted number of pairs (amongst those with complete genetic information) in which virus transmission occurred (‘‘transmission pairs’’) Npredicted” (Ssematimba et al., 2012, p. 4)

“part from the unknown and crisis organisation contacts, feed deliveries had the lowest per-contact probability of virus transmission of 0.0414 and potentially caused 2.63% of the new case farms while the egg transports had the highest per-contact probability of 0.308 and may have potentially caused 2.04% of the new case farms.” (Ssematimba et al., 2012, p. 4)

“The estimates were very similar for most of the exposure types. The only differences found, but these were small, were in the per-contact probabilities for the crisis organisation contacts for both alternative distributions and the other-professional contacts for only the unimodal distribution (see Table 2). For both alternative distributions, the probabilities per crisis organisation contact were within the 95% CI of the default distribution whereas for the unimodal distribution, the per otherprofessional contact probability reduced from 13.3% to 0.0%.” (Ssematimba et al., 2012, p. 5)

“With respect to the effect of the potential difference in tracing efforts on case and non-case farms – hence a possibility of underrepresentation of the contacts to non-case farms, we found that, with the worst tracing efforts, the contacts to case farms would be twice as likely to be traced as those to non-case farms.” (Ssematimba et al., 2012, p. 5)

“This implies that, at worst, the estimated probabilities could be double their ‘unbiased’ counterparts.” (Ssematimba et al., 2012, p. 5)

“In terms of per-contact risk, the estimates reveal that egg transports have the highest risk with approximately 31% chance of transmission followed by the rendering visits with a chance of transmission of 25%. The unknown contacts in the distance band of 0–1 km have the lowest risk per contact although, as is clear from the 95% confidence bounds, its estimated percontact probability is not significantly different from those of the other unknown contact categories.” (Ssematimba et al., 2012, p. 5)

“We expect that the implementation of preventive culling within 1 km of an infectious farm during the epidemic 5] has had a (strong) censoring effect on the detection of infected farms with 1 km of an infectious farm, thus producing a downward bias on the transmission probability per unknown contact within 1 km.” ([Ssematimba et al., 2012, p. 5)

“Our results suggest that, apart from the unknown contacts, egg delivery contacts are interesting targets for improvements in biosecurity due to their high per-contact probability (31%) in infecting the receiving farms. They further suggest that the biosecurity applied to the crisis organisation contacts seems to be adequate at least for preventing the persons themselves from becoming important fomites between registered visits. Overall, these findings provide a scientific basis to conduct further studies, epidemiological or otherwise, to evaluate the impact of improved biosecurity and minimized contact-frequency in controlling the between-farm spread of HPAI virus during epidemics.” (Ssematimba et al., 2012, p. 6)

“Virus by Professionals During Outbreak Control Activities” (Ssematimba et al., 2012, p. 7) .

Questions

• How can we leverage these data for different contexts?
• Can we apply these to different contexts outside of poultry farms?
• Could we use this type of survey to inform surveys for other livestock related outbreaks

Code/ Supplemental Data

In the article.

Implications for Infectious Diseases

Ssematimba A, Elbers ARW, Hagenaars TJ, Jong MCM de. Estimating the Per-Contact Probability of Infection by Highly Pathogenic Avian Influenza (H7N7) Virus during the 2003 Epidemic in The Netherlands. PLOS ONE 2012;7:e40929. https://doi.org/10.1371/journal.pone.0040929.

Related::

probability distribution medical risk factors genetic epidemiology genetics netherlands farms poultry chicken eggs

Imported: 2024-10-25

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Summary

Main Points

Abstract

A comprehensive literature review identifies 1415 species of infectious organism known to be pathogenic to humans, including 217 viruses and prions, 538 bacteria and rickettsia, 307 fungi, 66 protozoa and 287 helminths. Out of these, 868 (61%) are zoonotic, that is, they can be transmitted between humans and animals, and 175 pathogenic species are associated with diseases considered to be ‘emerging’. We test the hypothesis that zoonotic pathogens are more likely to be associated with emerging diseases than non-emerging ones. Out of the emerging pathogens, 132 (75%) are zoonotic, and overall, zoonotic pathogens are twice as likely to be associated with emerging diseases than non-zoonotic pathogens. However, the result varies among taxa, with protozoa and viruses particularly likely to emerge, and helminths particularly unlikely to do so, irrespective of their zoonotic status. No association between transmission route and emergence was found. This study represents the first quantitative analysis identifying risk factors for human disease emergence.

Notes

(5/3/2024, 5:57:55 PM)

“Infectious diseases account for 29 out of the 96 major causes of human morbidity and mortality listed by the World Health Organization and the World Bank (Murray & Lopez 1996) and 25% of global deaths (over 14 million deaths annually) (WHO 2000)” (Taylor et al., 2001, p. 983)

There could be more to do here.

Try again?

Can we have a few more notes? .

Questions

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Implications for Infectious Diseases

Imported: 2024-05-04 7:53 am

Taylor LH, Taylor LH, Latham SM, Woolhouse MEJ. Risk factors for human disease emergence. Philosophical Transactions of the Royal Society B 2001;356:983–9. https://doi.org/10.1098/rstb.2001.0888.

Related:: woolhouseRiskFactorsHuman2001

spillover emerging diseases emerging diseases emergence

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Summary

Woolhouse and colleagues examine some databases regarding zoonotic pathogens. They find that many pathogens that spill out into people are zoonotic and can cause human impacts.

Main Points

We typically under-sample environmentally transmitted pathogens. This includes fungi and other environmentally transmitted pathogens (e.g., bacteria and parasites). Helminths have a high likelihood of spilling over driven likely by exposure.

Abstract

A comprehensive literature review identifies 1415 species of infectious organism known to be pathogenic to humans, including 217 viruses and prions, 538 bacteria and rickettsia, 307 fungi, 66 protozoa and 287 helminths. Out of these, 868 (61%) are zoonotic, that is, they can be transmitted between humans and animals, and 175 pathogenic species are associated with diseases considered to be ‘emerging’. We test the hypothesis that zoonotic pathogens are more likely to be associated with emerging diseases than non–emerging ones. Out of the emerging pathogens, 132 (75%) are zoonotic, and overall, zoonotic pathogens are twice as likely to be associated with emerging diseases than non–zoonotic pathogens. However, the result varies among taxa, with protozoa and viruses particularly likely to emerge, and helminths particularly unlikely to do so, irrespective of their zoonotic status. No association between transmission route and emergence was found. This study represents the first quantitative analysis identifying risk factors for human disease emergence.

Notes

“Out of the emerging pathogens, 132 (75%) are zoonotic, and overall, zoonotic pathogens are twice as likely to be associated with emerging diseases than non-zoonotic pathogens.” (Woolhouse et al., 2001, p. 983)

“protozoa and viruses particularly likely to emerge, and helminths particularly unlikely to do so, irrespective of their zoonotic status” (Woolhouse et al., 2001, p. 983)

“We carry out such a test using the published literature to compile a list of organisms known to be pathogenic to humans, together with available information on whether they are zoonotic, whether they are regarded as emerging, and on their transmission routes and epidemiologies.” (Woolhouse et al., 2001, p. 983)

“Although the de¢nition of species is di¤cult for some infectious organisms, this is the most appropriate level of classi¢cation for the vast majority of pathogens and avoids biases that would otherwise be introduced by organisms that exhibit a large amount of subspeci¢c variation (e.g. some species of Salmonella and Listeria).” (Woolhouse et al., 2001, p. 984)

“To appear in the database a species name must ¢rst have appeared in an up-to-date source text (published within the last ten years), and second appeared in an upto-date nomenclatural reference source, where available (see above), or appeared in a second up-to-date source text, or appeared in an ISI Web of Science Citation Index search of the last 10 years.” (Woolhouse et al., 2001, p. 984)

“A genus was considered to be zoonotic, and/or emerging, and/or transmissible by a particular route if at least one species in it had that characteristic.” (Woolhouse et al., 2001, p. 984)

“Analyses were performed comparing emerging and non-emerging species by taxonomic division, transmission route and zoonotic status and by combinations of these characteristics.” (Woolhouse et al., 2001, p. 984)

“Overall, 19% are viruses or prions, 31% are bacteria or rickettsia, 13% are fungi, 5% are protozoa, and 32% are helminths. Thirty-¢ve per cent of zoonotic pathogens can be transmitted by direct contact,” (Woolhouse et al., 2001, p. 985)

“61% by indirect contact, 22% by vectors, and for 6% the transmission route is not known.” (Woolhouse et al., 2001, p. 985)

“The clearest patterns are that helminths are overrepresented among zoonoses and that fungi are underrepresented.” (Woolhouse et al., 2001, p. 985)

“zoonoses are more likely to be transmitted by indirect contact or by vectors, and are less likely to be transmitted by direct contact when compared with all pathogens” (Woolhouse et al., 2001, p. 985) have to consider some level of detection bias here. if fungi are underrepresented then we would expect environmental transmission to be underestimated in the literature

“This is substantially more than expected if zoonotic and nonzoonotic species were equally likely to emerge, and corresponds to a relative risk of 1.93.” (Woolhouse et al., 2001, p. 986)

“The majority of pathogen species causing disease in humans are zoonotic (868 species, i.e. 61% of the total; electronic Appendix A).” (Woolhouse et al., 2001, p. 986)

“In addition, zoonoses are relatively likely to be transmitted indirectly (including transmission by intermediate hosts) or by vectors, suggesting that these transmission routes may be associated with lower host speci¢city (Woolhouse et al. 2001).” (Woolhouse et al., 2001, p. 987)

“The observation that the route of transmission of over 200 human pathogens (both zoonotic and non-zoonotic) remains unknown emphasizes the need for improved understanding of the biology of infectious agents in general.” (Woolhouse et al., 2001, p. 987)

“Human-to-human transmissibility is a risk factor for emergence across all pathogens, with a relative risk of 2.60.” (Woolhouse et al., 2001, p. 987)

“Nonetheless, we anticipate that pathogen biology also contributes to the likelihood of emergence, including such factors as genetic diversity, generation time and existence of a reservoir (whether zoonotic or environmental).” (Woolhouse et al., 2001, p. 987)

“Nonetheless, animal and human diseases can be closely associated; recent examples include Rift Valley fever in Kenya and Somalia (WHO Press Release 1998), Nipah virus in Malaysia and Singapore (Chua et al. 2000), West Nile virus in the United States (Lanciotti et al. 1999) and Hendra virus in Australia (Westbury 2000).” (Woolhouse et al., 2001, p. 987) .

Questions

Code/ Supplemental Data

Implications for Infectious Diseases

Imported: 2024-05-04 2:16 pm

Woolhouse MEJ, Dye C, Taylor LH, Latham SM, woolhouse MEJ. Risk factors for human disease emergence. Philosophical Transactions of the Royal Society of London Series B: Biological Sciences 2001;356:983–9. https://doi.org/10.1098/rstb.2001.0888.

Related:: taylorRiskFactorsHuman2001

epidemiology zoonoses public health risk factors emerging diseases

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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

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Summary

In this article, we extent the literature on the evolutionary effects of variable migration and investigate how temporal variation in immigration rates influences adaption to environments outside a species’ niche (i.e., sink environments).

When immigration events are too frequent, gene flow can hamper local adaptation in sexual species, but sufficiently infrequent pulses of immigration allow for repeated opportunities for adaptation with temporary escapes from gene flow during which local selection is unleashed.

We address how the frequency of immigration pulses influences adaptation to sink environments…. we will also examine how density dependence and genetic architecture modulate the impact of pulsed immigration on local adaption. We conjecture that changing the frequency of immigration could alter the relative impacts of the opposing positive (e.g., increased genetic variant) and negative (e.g., maladaptive gene flow) effects of immigration sketched above.

“we observed an adaptive benefit of spacing out immigration events only when sink environments were sufficiently harsh (high v). For relatively mild sink environments, the probability of adaptation was high for all immigration frequencies (fig. 3B, asterisks), but for harsher sink environments the probability of adaptation was low for constant immigration but then increased as immigration events became less frequent (fig. 3B, circles and crosses).” (Peniston et al., 2019, p. 325)

More generally, immigration frequency has some impact as immigration adds diversity of genes to a sink, but if it is too frequent, local adaption cannot occur because the population of maladapted organisms is too high (this is similar to findings in Virulence on the Edge). However, if there is a balance between immigration rates and local adaptation, then evolution is more likely to occur. This varies between the harshness of the sink where with mild sinks, immigration cohorts could more readily adapt, while in more harsh sinks the less frequent pulses led to local adaptation.

Methods

Will use

1. Deterministic models
2. Stochastic individual-based simulations
   1. Deterministic models don’t allow for extinction
   2. Deterministic models assume that the population if fixed
   3. Similalrly ignores stochastic effects

• Without density dependence
• Genetic architectures
  • Single-locus haploid
  • Single-locus diploid
  • Multi-locus quantitative genetic variation
• One way sink (i.e., “black hole”) with discrete generations
• Fitness was the product of a phenotype-dependent probability of survival until adulthood and a phenotype independent fixed fecundity
• Immigrants arrive as adults

$$\begin{align} N = N_1 + N_2 \newline W_i(N) = \text{Density depdent abskyte fitness of genotype } A_i \newline V_i(N) B= \text{Absolute fitness} \end{align}$$ For a sink

$$\begin{align} W_2(N) \lt 1 \ \text{for all N} \newline W_1(N) \gt 1 \ \text{for some N} \end{align}$$

Haploid model

Then the mean fitness, $\bar W$ is:

$$\begin{align} p = \frac{N_1}{N_1+N_2} \newline \bar W(N) = pW_1(N) + (1-p) W_n(N) \newline W_i(N) = \frac{w_i}{1+CN} \end{align}$$ Where if c = 0, then there is no density dependence.

Diploid model

$$\begin{align} \bar W (N) = p^2W_{11} + 2p(1-p)W_{12}(N) + (1-p)^2W_{22}(N) \newline W_{ij}(N) = \frac{w_{ij}}{1+cN} \end{align}$$

Main Points

• Frequency of immigration events did not affect adaption to a sink environment

Introduction

Immigrations

• Increases genetic diversity
• Boosts the population size
• Increase adaption by counteracting Allee effects vocabulary/Allee

Questions

Code/ Supplemental Data

Lots of C++ data are available.

Implications for Infectious Diseases

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Summary

Main Points

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Summary

Main Points

• Species coexistence is driven by the interplay between interspecies competition and intraspecies competition. As long as intraspecies competition is greater than interspecies competition the conditions exist for an equilibria which includes coexistence.

Environmental Dimensionality

Below we show the per capita abundance investigating the role of different relationships with inter and intra species competition. Here $a$ is taken as some growth rate for species 1 and 2 which is reduced by some additive factor based on the intraspecies competition (e.g., $a_{11}$) and the interspecies competition (e.g., $a{12}$).

$$\begin{align} \frac{1}{N1} \frac{dN_1}{dt} = a_1 - a_{11} N_1 - a_{12}N2 \newline \frac{1}{N2} \frac{dN_2}{dt} = a_2 - a_{22} N_2 - a_{21}N1 \end{align}$$

[!tip] Note that this can be placed into matrix notation and that matrix will have a range of two. The rank of this matrix tells us the dimensionality of the system as well as the dimensionality of the environment

Questions

Code/ Supplemental Data

Implications for Infectious Diseases

Summary

Under what conditions can species coexist? Species is the product of the interaction and dynamics of populations within the context of other species, predators, and resource availability.

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Summary

Main Points

Questions

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Summary

Main Points

If infections are localized, spatial separation increases the degrees of freedom of a host-parasite system permitting a rich array of dynamical behaviours to arise even in a spatially homogenous world (e.g., Hassell et al. 1994)

Local populations might have different carrying capacities or involve anisotropic flows.

This leads to the potential for asymmetries among habitats in the degree of the impact of spatial coupling on local ecological and evolutionary dynamics

Sink populations may readily arise at the edges of species’ ranges, or where habitats that differ greatly are juxtaposed. Given that the genetic variation is present, natural selection might be expected to improve the ability of a species to utilize the sink habitat.

However, there appear to be severe constraints on the ability to have adaptive evolution in sinks (i.e., adaptive evolution to the conditions within the sinks).

leading to a king of evolutionary conservatism in spatially heterogeneous environments.

Yet few studies (of gene flow and selection, md) focus explicitly on the potential implications of source-sink dynamics for our understanding of the evolution of virulence and resistance (for an analysis of comparable issues in predator-prey coevolution along a gradient, see Hochberg and Van Baalen 1998).

Motivating examples

1. Generate some source sink scenarios
2. Develop potential exploratory models for these scenarios and discuss the initial stages of adaption for the host-pathogen interaction in a sink
3. Generalize some conclusions given these simple models

Motivating examples

1. Organic farmer growing corn and sees a fungal blight that reduces crop yield. The question is how much seed to hold back (which might have resistance to this blight).
2. Animal husbandry with a benign pathogen in the animals. What concern should there be regarding a more serious illness emerging.
3. Doctor in a nursery ward with the emergence of nonsocomial pathogen

What unifies these three situations is that they all involve spatial dynamics (in the broad sense of mixing together individuals drawn from distinct populations); depending upon the quantitative details, these scenarios could involve a source-sink structure for either the host of the pathogen

Models of source-sink dynamics

Several simplifying assumptions are made for modelling these dynamics

1. Host exhibits continuous generations and any direct density-dependence is dominated by the effects of the pathogen
2. Once a host is infected so that it is infectious, it cannot be super- or multiply- infected and infectious hosts do not recover/give birth and remain infectious until removed from the population
3. These assumptions mean that co-circulation does not occur

With no potential for host recover, host evolution in response to the pathogen is related to the likelihood of successful infection in the first place and to the production of infected host individuals who themselves are infectious. The transmission parameter, $\beta$, at the heart of a standard epidemiological model defines the rate at which susceptible hosts themselves become infective.

Selection on the host tends to reduce $\beta$ (i.e., selection favors reducing transmission of the pathogen from infectious host to susceptible host).

Some additional definitions in this context:

$S$: the density of immigrant (ancestral) host in the sink habitat $S_{mut}$: the density of a novel type of host $I$ : the density of infected hosts Virulence: the average fitness across all hosts carrying the pathogen, including those whose successful defenses are reducing the pathogen titer toward extinction.

Model 1 Host evolution in a sink with generalist pathogen

The source contains the host alone whereas the sink has both the pathogen and the host. The pathogen is a generalist, so its dynamics are decoupled from from the focal host (e.g., no evolution in this sink). Additionally, assume that density of infected hosts is fixed at I. Here we are not concerned with pathogen persistance. We also have some host immigration (from the source) at some value, $H$. Evolution of virulence is governed by evolution in $\beta$ in the case.

We have some intrinsic growth rate $r$ which is government by births, $b$ and deaths $b$.

$$\begin{align} r = b - d \newline \frac{dS}{dt} = rS - \beta S I + H \end{align}$$

Important conditions

1. If $r \lt 0$ then this habitat is an intrinsic sink, i.e., in the long run, the population will go extinct regardless of the presence of the pathogen.
2. If $0 \lt r \lt \beta I$ then the presence of the pathogen can cause the habitat to become a sink (i.e. an induced sink). The potential for adaptive evolution of the resistance by hosts is different in these intrinsic sinks is different here.

Now adding evolution

$$\frac{dS_{mut}}{dt} = r_{mut} S_{mut} - \beta_{mut} S_{mut} I$$

1. In the case where the habitat is an intrinsic sink (r < 0) then we will have Error in rendering LaTeX so even it is a sink without pressure, then intrinsic sinks do not benefit from natural selection selecting for alleles that confer host resistance (even if they are cost-free). Likelihood of local adaptation in intrinsic sinks via locally favoured mutations is reduced in an intrinsic sink
2. However, in an induced sink, a novel allele can increase in abundance if $r \gt \beta_{mut} I$ with a minimum magnitude of the mutational effect required as $\Delta \beta = \beta - r/ I$. If there is a fitness cost then it must be such that $\Delta \beta$ is exceeded even if $r_{mut} \lt r$

Three main conclusions:

1. Local adaptation to of the host to the pathogen is less likely if productivity in the sink is low, or pressure from the pathogen (I) is high.
2. At larger I, or smaller r, evolution occurs only if mutants with a sufficiently large effect arise
3. We do not expect evolution in the hosts in intrinsic hosts at all

Immigration can increase the local population size and make it more likely that mutants will arise. More immigration also means more variation.

Model 2 Host evolution in a sink with specialist pathogen

Host flows from the sink with a specialist pathogen with a dynamical response such that the infection level depends on host dynamics. This takes the form of the classic SI model with host immigration, H, also included and where some pathogenicity from the infection is also included as, $\alpha$, and where $\mu$ represents the addition of the intrinsic death rate and this pathogen induced death rate.

$$\begin{align} \frac{dS}{dt} = r S - \beta S I + H \newline \frac{dI}{dt} = \beta S I - \mu I \newline \mu = d + \alpha \end{align}$$

Then we have the following equilibrium points:

$$\begin{align} S^{\star} = \frac{\mu}{\beta} \newline I^\star = \frac{r}{\beta} + \frac{H}{\mu} \end{align}$$

Thus if r <0 then $H \gt |r| d/ \beta$ for pathogen persistance meaning that a specialist pathogen must be sufficiently transmissible to persist in an environment that is an intrinsic sink for its host. Additionally, there must be enough hosts coming in to sustain the sink.

The abundance of infected individuals rises until the negative growth rate of the host population just matches the rate of input (immigration) from outside.

We can then re-introduce the evolutionary component:

$$\frac{dS_{mut}}{dt} = r_{mut} S_{mut} - \beta_{mut} S_{mut} I$$

Again, if the habitat is an intrinsic sink for the host ($r_{mut} \lt 0$) then evolution does not promote local adaptation by the host (as $S_{mut}$ also tends to 0). However, if the habitat is an induced sink:

$$\begin{align} \frac{dS_{mut}}{dt} \gt 0 \newline iff: \newline \beta_{mut} \lt \beta \big(1 + \frac{H \beta}{r \mu} \big) \end{align}$$

Thus $r$ and $H$ have opposing effects on local adaptation as

• increasing r increases the range of mutational effects on $\beta$
• Increase $H$ makes adaption more difficult ![[Pasted image 20230904135426.png]]

So key conclusion, local adaptation towards lower transmission rates is more likely for:

1. Habitats with high host productivity (and unlikely in intrinsic sinks)
2. Habitats in which the pathogen initially has a low rate of transmission ($\beta$)
3. Habitats in which infected individuals are short-lived (e.g., high $\mu$)
4. Unproductive source habitats (low H)

Host immigration indirectly increases pathogen abundance and thus increases the mutational effect required to allow the conditions for the lower transmission rate population to persist. Including the cost function ($r_{mut} \lt r$) makes local adaption arrise (See Hochberg and Van Baalen 1998)

Model 3 Host evolution in a sink with specialist pathogen

Habitat contains both host and pathogens and there is cross-habitat infection with infected individuals in the source infecting healthy individuals in the sink. This is comparable to the notion of disease reservoirs as described by May and Anderson (1991).

$$\frac{dI}{dt} = \beta_{source} I_{source} S + \beta I S - \mu I$$

Note that $\beta_{source} I_{source} S$ reflects the force of infection on health, susceptible hosts in the sink caused by infected individuals in the source.

If we assume that $I_{source}$ is constant for the sink and that S is the host carrying capacity, K, then we can make some simplifying assumptions. Thus this habitat will be an instrinsic sink if:

$$\begin{align} \beta K \lt \mu \newline \text{OR} \newline \beta \lt \frac{\mu}{K} \end{align}$$

Factors constributing to being an intrinsic sink for the pathogen

• Host is scarce (low K)
• Transmission rates are low (low $\beta$)
• Infected hosts have high death rates either because hosts have instrinsically high deaths rates (high d) or pathogen is highly virulent (high $\alpha$)

However, because of external inputs, a pathogen can persist in an intrinsic sink:

$$\frac{dI_{mut}}{dt} = \beta_{mut} I_{mut} K - \mu_{mut} I_{mut}$$

Thus a novel strain can survive if $\beta_{mut} \gt \mu_{mut} / K$. Note that if the transmissibility increases it will be selected for if the impact on virulence is not too high.

A novel mutation will be successful if $\mu = \mu_{mut}$ then a novel mutation will be successful if $\Delta \beta = \mu / K -\beta$ . Evolution in a host habitat that is a sink for the pathogen thus occurs only if mutants have large effect upon transmission arise.

In the case where there is a trade-off between mortality and transmissibility, the picture is a little more complicated. Higher values of K (more population abundance) will allow for higher degrees of total mortality by expanding the adapability range into the fitness space.

Thus pathogen is sustained by immigration which is a pathogen sink, then local adaption in rates of transmission and virulence are more likely if:

• The host population is abundant
• The immigrant parasite strain is not too badly maladapted to the local host in the first place

A more virulent pathogen strain is most likely to invade a sink habitat when:

• Mutations occur that confer a substantial difference between the immigrant and mutant pathogen strains. Small changes are less likely to become locally adapted to hosts in the sink
• Host density in the sink, K, is high. (with increasing K, the slope of the dashed line decreases increasing the feasible invasion space for more virulent pathogens)
• The pathogen in the immigrant stream is low in virulence to begin with

A simple message of the above models is that the demographic context of the initial stages of contact with novel hosts may be crucial in predicting emergence. IF the demographic context is that the novel hosts are sinks for the pathogens invading from another species, then adaptation by the pathogen to the host requires mutants of large effect; if such genes are rare, the emergence of the disease as a serious problem may be unlikely

Questions

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Implications for Infectious Diseases

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Summary

In this article Perron and colleagues experimentally tested the impact of immigration and environment harshness on Pseudomonas aeruginosa. In their full factoral experiment they tested how the immigration rate of the bacteria as well as the treatment environment could impact the fitness exploring the idea behind source-sink dynamics. They found that:

• Increasing immigration rates had a positive impact on the density of Pseudomonas in all environments (in alignment with source-sink dynamics theory)
• Antibiotic therapy introduced mutations with fitness costs (i.e., fitness costs were introduced in order for the bacteria to reproduce within harsh environment)
• So called bi-therapy (or simultaneous multi-drug therapy) induced higher fitness costs
• Cycling of different antibiotics imposed fitness costs, but less so than the bi-therapy.

This study is important for a few reasons:

• Experimental support for source-sink dynamics (and associated immigration) on introducing more rapid evolution of resistance
• Important implications for the control of nosocomial infections as immigration of pathogens (especially under monotherapy)

this means that in important nosocomial infections, such as those by P. aeruginosa, the immigration of susceptible bacteria established in an antibiotic-free reservoir (e.g. contaminated water; Trautmann et al. 2005) into transient secondary niches supplemented with antibiotic (e.g. respiratory tract of treated patient; Festini et al. 2006) can not only foster the rapid evolution of antibiotic resistance, but can also create resistant mutants with little or no fitness cost.

Main Points

The evolution and maintenance of antibiotic resistance depend on both the probability of resistance mutations and the pleiotropic fitness cost associated with resistance (Andersson 2006); the latter often manifested as slower growth rate or a reduced competitive ability in the absence of antibiotics (Andersson 2003)

There are some concerns regarding epistasis in that the harsher environments will result in fixed mutations or evolutionary pathways in which a local, but suboptimal, evolutionary equilibrium could exist.

As with the probability of resistance evolution, the fitness of a resistant mutant that goes to fixation will be a positive function of the mutation supply (Levin et al. 2000) rate, for which migration is likely to be a key determinant as higher migration brings more mutants. Under higher migration rates, many resistance mutations will arise simultaneously, allowing selection to fix the mutation with the lowest fitness cost. By contrast, if mutation supply rate is low, the less fit resistant mutation is likely to reach fixation before a better mutation appears.

Source-sink models

These models have been applied more recently to micro-organisms to explain the evolution of virulence (Sokurenko et al. 2006; Chattopadhyay et al. 2007)

source - population is found in its “fundamental niche” which is a set of environment al conditions and resources that permit a population to persist, grow (i.e., birth rate larger than death rate over some range of densities), and produce immigrants (Hutchinson 1978)

sink- a population fundamental niche (a harsh environment) has a mean fitness lower than 1 (e.g. death rate exceeds the birth rate) and cannot be sustained without passive (Holt 1985/10) or active (Pulliam 1988) immigration and is therefore referred to as a ‘sink’ population

With sustained immigration, there should be adaption to the sinks and that adaptation will occur quicker in less harsh environment. Adaptation will then look often like punctuated and rapid growth from this interplay of immigration and selection.

Conversely, immigration can also constrain adaptive evolution, because gene flow can swamp locally favoured variants, and because immigrants can compete with better-adapted residents. This effect hinders adaptation by maintaining the population away from the local fitness optimum.

![[Pasted image 20230831120243.png]]

Questions

The statistics were a bit wild and I wish we had access to the original experimental data. We could likely calculate growth rates under the different conditions rather than including a second order polynomial as they have described in their methods.

Code/ Supplemental Data

Connections to infection diseases

Terms

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Summary

Main Points

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Implications for Infectious Diseases## Summary

Traditionally we have examined the “strength” of an epidemic as measured by $R_0$. This typically minimizes the role that the related speed, r, can play during an epidemic. In this analysis Duschoff and Park examine the how thinking about strategies on invention can be more intuitive understood through diminishing the strength of an epidemic (e.g., mass vaccination, lockdown, condom use) versus the speed of an epidemic (e.g., contract testing, quarantines, etc.).

Main Points

• Speed and strength are related (e.g., $R_0$ and $r$ are related through the generation time, so the conclusions will be similar, but some of the nuance is better captured through this framework).
• Strength-based interventions are clearer for the interventions that target entire populations
• Speed-based interventions are clearer for interventions which target infected people
• Early in epidemics you may only really have $r$ because the other parameters are unknown or more difficult to estimate

Questions

Code/ Supplemental Data

https://github.com/mac-theobio/Speed_and_strength

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Notes from the reading

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