Epizootics Of The Plague In Priarie Dog Populations

Published: May 23, 2018

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Article

In the following pages, we will be discussing an by Colleen Webb, Christopher Brooks, et al, about the transmission of the plague in prairie dog towns ^[1].

Executive Summary

In this study, after collecting data in the field as well as from previous literature, numerical and sensitivity analysis is performed on a population of prairie dogs in the Pawnee National Grasslands () in northern Colorado. The authors analytically and numerically succeed in demonstrating that the plague is spread through a town of these small mammals not by the previously believed "blocked fleas", but by other factors. The authors discover there may be several other factors driving the epizootics that are much more likely to cause extinction of a local population.

Background

"Yersinia pestis" is the bacterium known to cause three different plagues: pneumonic, septicemic, and bubonic, to which both animals and humans are susceptible. These plagues are known to be responsible for highly contagious and deadly epidemics, including the Black Death, which destroyed approximately a third of the European population between the years 1347 and 1353. ^[2].

It has been long known that the (Cynomys ludovicianus) is one of the rodent populations most susceptible to the plague (). Because of its high susceptibility, if an infected host is introduced into a "town" of these mammals, frequently, an follows in the population, resulting in a 98% mortality rate (based on the stochastic models utilized in this article). Another advantage to researching this particular species is the fact that it is vulnerable to all forms of transmission of the bacteria.

For many years, researchers believed that the mass deaths in prairie dog towns were mainly caused by the bites of "blocked" fleas; or fleas which, after contracting the disease, form a blockage in their midguts, resulting in starvation that leads to an aggressive feeding behavior and frequent regurgitation (in attempt to rid themselves of the blockage). The authors of this article disprove this common misconception, demonstrating with a thorough sensitivity analysis of the observed population, that these "blocked" fleas are not nearly as important in driving the epizootic as some other factors. There are two other ways of transmitting the disease: through airborne methods and contact with a "short-term infectious reservoir." This reservoir is a combination of deceased prairie dogs carcasses, waste from infected prairie dogs and possibly other smaller mammals which live in the vicinity of the prairie dog colony. Another alternative is that there exist certain species of insects that do not form a "blockage", but in fact carry the bacterium on their mouth parts. This article shows that the spread of Y. pestis results from one or a combination of these two alternative transmission routes, and their analysis helps explain how and why this occurs.

History

Susceptible-Infected-Removed () models are widely used in modeling the spread of infectious diseases. Depending on the individual topic being researched, parameter values will vary and the dynamics of the system can vary as well. The simplest model contains a susceptible class which moves to the infected class at a certain rate based on the density of the susceptibles and the infected, an infected class that can move to either the removed or the susceptible class, and a removed class which can possibly become part of the susceptible class. This model is very basic and while informative, brushes over many dynamics that a system can also exhibit.

The model under consideration has the same basic principles, but has been extended to more clearly explain the behavior of the system. Instead of only one species being considered, another species is introduced that helps spread the disease. Also, the prairie dogs themselves are broken up further into classes: susceptibles, the exposed class which grows in accordance to the amount of fleas that are "questing" for food, an infected class and a reservoir class. Ignoring the dynamics of the fleas, only an exposed class has been added. Also, the removed class consists of the deceased prairie dogs and other factors which contribute to the spread of the disease as well as the infected class. Thus, in this case, the "removed" class can also infect the susceptible and the exposed classes.

Mathematical Model

Deterministic Model

Parameter Values

There are multiple parameters for the model used in this article. They were derived to best represent actual parameters of prairie dog populations from previous literature on the same topic as well as from the data obtained in the field. Note that time units are in days.

Parameter	Value	Description
$r$	0.0866	Intrinsic rate of increase (host)
$K$	200	Carrying capacity (host)
$\mu$	0.0002	Natural mortality rate (host)
$\beta_F$	0.09	Blocked vector transmission rate
$\beta_C$	0.073	Airborne transmission rate
$\beta_R$	0.073	Transmission rate from reservoir*
$B$	20	Number of burrows host enters**
$\sigma$	0.21	1 / Exposed period (host)
$\alpha_F$	0.5	Blocked vector mortality rate
$\alpha_C$	0.5	Contact mortality rate**
$\lambda$	0.006	Reservoir decay rates
$\delta$	0.05	Rate of leaving hosts
$a$	0.004	Searching efficiency of questing fleas
$\mu_F$	0.07	Natural mortality rate (vector)
$r_F$	1.5	Conversion efficiency (vector)
$\gamma$	0.28	Transmission rate: hosts to vector
$\tau$	0.009	1 / Exposed period (vector)
$s$	0.33	Disease induced mortality rate (vector)

*Assumed to be the same as airborne transmission

**Estimated from field data

Research by the authors in the Pawnee National Grasslands in Northern Colorado and literature from other authors led to the estimates of most parameters in the model. All of the parameters that do not rely on the bacteria were obtained through this research as well as other statistics on prairie dog and flea dynamics. For the disease parameters, laboratory experiments were used with the bacterium Y. Pestis and O. Hirsuta, which portrays similar dynamics with Y. Pestis. For some of the transmission parameters, estimates were also taken from observations and data obtained from similar species of mammals like the California vole (Microtus Californicus).

Differential Equations

The differential equations used are:

The host model:

${\frac {dS}{dt}}=rS(1-N/K)-{\frac {\beta _{{C}}S(I_{{C}}+I_{{F}})}{N}}-\beta _{{F}}F_{{IQ}}S(1-e^{{-aN/\beta }})-\beta _{{R}}S{\frac {M}{B}}-\mu S,$

${\frac {dE_{{F}}}{dt}}=\beta _{{F}}F_{{IQ}}S(1-e^{{-aN/B}})-E_{{F}}(\sigma +\mu ),$

${\frac {dE_{{C}}}{dt}}=\beta _{{R}}S{\frac {M}{B}}+{\frac {\beta _{{C}}S(I_{{C}}+I_{{F}})}{N}}-E_{{C}}(\sigma +\mu ),$

${\frac {dI_{{F}}}{dt}}=\sigma E_{{F}}-I_{{F}}\alpha _{{F}},$

${\frac {dI_{{C}}}{dt}}=\sigma E_{{C}}-I_{{C}}\alpha _{{C}},$

${\frac {dM}{dt}}=\alpha _{{C}}I_{{C}}+\alpha _{{F}}I_{{F}}-\lambda M.$

The vector submodel:

${\frac {dF_{{SQ}}}{dt}}=\delta F_{{SH}}+r_{{F}}F_{{O}}({\frac {N}{1+N+F_{0}}})-F_{{SQ}}[\mu _{{F}}+(1-e^{{-aN/B}})],$

${\frac {dF_{{SH}}}{dt}}=F_{{SQ}}(1-e^{{-aN/B}})-F_{{SH}}[\mu _{{F}}+\delta +\gamma ({\frac {I_{{C}}+I_{{F}}}{N}})],$

${\frac {F_{{EQ}}}{dt}}=\delta F_{{EH}}-F_{{EQ}}[(1-e^{{-aN/B}})+\tau +\mu _{{F}}],$

${\frac {F_{{EH}}}{dt}}=F_{{SH}}\gamma ({\frac {I_{{C}}+I_{{F}}}{N}})+F_{{EQ}}(1-e^{{-aN/B}})-F_{{EH}}(\tau +\mu _{{F}}+\delta ),$

${\frac {dF_{{IQ}}}{dt}}=\delta F_{{IH}}+\tau F_{{EQ}}-F_{{IQ}}[s+\mu _{{F}}+(1-e^{{-aN/B}})],$

${\frac {dF_{{IH}}}{dt}}=F_{{IQ}}(1-e^{{-aN/B}})+\tau F_{{EH}}-F_{{IH}}(s+\mu _{{F}}+\delta ).$

In this model, the prairie dog population are subdivided into six classes: susceptibles, $S$; exposed, $E_F$, and infectious, $I_F$听by contamination by the blocked fleas; and those exposed, $E_C$, and infectious $I_C$听through direct contact with the reservoir, $M$.听From this, the total number of prairie dogs is described by $N=S+E_F+E_C+I_F+I_C$.

Similarly, the fleas are divided into six classes: susceptible and questing,$F_{SQ}$, susceptible and on the host,$F_{SH}$, exposed and questing,$F_{EQ}$, exposed and on the host, $F_{EH}$, infectious and questing,$F_{IQ}$, and infectious and on the host, $F_{IH}$.

These ODES use probability- and density-dependent contact between different groups, depending on the characteristics of each (i.e. questing fleas and on-host fleas exhibit different types of growth and death). It also seems that prairie dog colonies are usually structured in groups, which affect how diseases are spread throughout the population. Depending on the proximity of different family groups, the transmission rates between prairie dogs can vary. Because of the more random-characteristics of flea transmission, dynamics resulting from fleas are modeled primarily using frequency-dependent methods. To model the transmission caused by the short-term reservoir, density-dependent methods are used. As more prairie dogs die, the population becomes mixed as the structure of the colonies breaks down. Thus, density-dependent methods are more appropriate.

In the prairie dog classes, mortality rates vary due to biological differences. The prairie dog mortality rate, $\mu$, only affects the susceptible class and the exposed class due to the average length of time from exposure to death from infection is around 2 days. The number of holes that any prairie dog will enter, $B$听is used to estimate area of the prairie dog colony. The amount of time that a prairie dog will remain in the exposed class before it moves to the infected class is given by $\sigma^{-1}$. Thus, $\sigma$听indicates the rate at which the exposed class moves to the infected class. Also, the infected class contributes to the short-term reservoir proportional to its density by the parameter $\lambda$.

The death rate of fleas is given by the parameter $\mu_F$听and the rate at which the fleas die due to blockage is given by the parameter $s$. The transition from on-host fleas to questing fleas is given by the parameter $\delta$听and the function $1-e^{\frac{iaN}{B}}$. Thus, the transition from questing to on-host classes is based on the number of prairie dogs, $N$, the proximity of other burrows to the current one, B, and the efficiency of the fleas in finding a new host, a. Due to the short infection time and the abundant amount of blood needed to reproduce, reproduction of fleas is restricted completely to the on-host class. Since $\tau^{-1}$听is the time is takes for the fleas' proventriculus to get blocked, the rate $\tau$听indicates the rate at which exposed fleas become able to infect the host due to regurgitation after the blockage.

Sensitivity Analysis

Sensitivity analysis was used to find which parameter values had the greatest impact on changing the extinction probability using the following equation:

$\Sigma \approx {\frac {ln(V(P)-ln(V(P_{{0}}))}{ln(P)-ln(P_{{0}}))}}.$

This average sensitivity was calculated based on over 1000 simulations. The extinction probability is given by the function $V$. $P_0$听is the default parameter whose sensitivity is being analyzed and $P$听is another arbitrary parameter value.

Stochastic Model

The stochastic model is fairly similar to the deterministic model. The only difference is that if the prairie dogs die out (i.e. N = 0), the rate at which infected and exposed classes of fleas grow due to contact with the prairie dogs will become zero. The assumptions of the stochastic model are as follows: events occur only one at a time, all events occur independently of any other event and that the probability of an event occurring per unit time is held constant.听