Ordinary Differential Equation Model for the DiseaseSpread of Kuru within the Fore Tribe of Papua NewGuineaHarrison Tuckman1,*,+ and Kylee Gilman2,**,+

1College of William and Mary, B.S. in Mathematical Biology, Williamsburg, VA 23187, US2College of William and Mary, B.S. in Applied Mathematics, Williamsburg, VA 23187, US*hgtuckman@email.wm.edu**kjgilman@email.wm.edu+these authors contributed equally to this work

ABSTRACT

Kuru is a neurodegenerative prion disease which was common among the Fore people of Papua New Guinea.

Prions are misfolded proteins that cause other proteins to misfold, eventually causing fatal neurodegeneration.

Prions are transmitted in a variety of ways, such as by oral consumption, which is relevant to the Fore people who

frequently perform endocannibalistic cultural practices. The goal of this paper is to effectively model the spread of

kuru. In order to model kuru, a modified S� I�R was used in which the R population originally meant for recovered

individuals was replaced by a Di population, representing dead individuals who had kuru. From this model, a

simpler S� I model was constructed and analyzed. According to the model, the infection rate of kuru and the death

rate of individuals from kuru prove to be highly influential in the behavior of the model. Once the death rate by kuru

exceeds a critical value, a fixed point emerges in the biologically relevant portion of the system. In addition, as the

death rate by kuru increases beyond this critical value, the infected population size of the fixed point decreases.

Furthermore, as the infection rate of kuru increases, the population stabilizes with a smaller susceptible population.

IntroductionKuru is a neurodegenerative prion disease that was common in the Fore tribe in Papua New Guinea. In order tounderstand the effects of the disease and how the disease is transmitted, one must understand what prions are, andthe history and cultural practices of the Fore people.

The Molecular Biology of PrionsPrions are misfolded proteins that cause other similar proteins to misfold, leading to neurodegenerative disease andeventually, death. Common prion diseases in humans include sporadic Creutzfeldt-Jakob disease (sCJD), iatrogenicCreutzfeldt-Jakob disease (iCJD), and kuru. However, prions are not limited to humans. For example, bovinespongiform encephalopathy, also more commonly known as Mad Cow disease, is the result of a prion disease similarto that of CJD [1].

Acquisition of Prions

Prions are acquired in a number of ways. For example, prions can be acquired spontaneously, as in sporadicCruetzfeldt-Jakob disease, by infection, as in iatrogenic Cruetzfeldt-Jakob disease, or genetically, as in Gerstmann-Straussler-Scheinker disease [1]. Additionally, in a laboratory setting it was shown that prion diseases are transmissi-ble by oral consumption [2]. In the case of kuru, oral consumption plays the largest factor in transmission.

Aggregation of Prions

After a prion enters a healthy individual, it causes other proteins to misfold and take the same incorrect shape. Whatmakes prions interesting is that they contain no nucleic acids or any form of genetic information [3]. While virusesand most other infectious agents rely on having cells read genetic information in order to produce infectious proteins,

prions rely solely on their structure to cause other normal proteins to misfold [4]. It is believed that the tertiarystructure of the prion protein interacts with the normal protein in a post-translational process which causes thenormal protein to misfold and attain a high beta pleated sheet concentration [3], though little is known about howthis process actually works or occurs. After enough prions have been formed, they begin to aggregate in the centralnervous system. At that point, the prions begin to cause neuronal death and damage, though it is not understood howthis death occurs [1]. Eventually, after enough neuronal death and degeneration has occurred, the individual dies.There is currently no treatment or cure for prion diseases, and the body has no way of curing itself of the disease,however there has been some recent experimentation using antibodies to inhibit and delay the replication of prions[5].

The History and Culture of Kuru within the Fore people of Papua New GuineaKuru was transmitted through endocannibalistic rituals within the Fore tribe of Papua New Guinea. In addition, anyneighbors with whom the Fore people intermarried were susceptible to the disease. In all of kuru’s history, it onlygreatly affected these groups of people. [6].

Historical Presence of Kuru

The Fore reported that kuru first entered their tribe around 1920 [7]; kuru then peaked in the 1950s when it wasfinally studied [8]. With the light that scientific studies brought to the endemic, kuru has steadily decreased inoccurrence due to government intervention effectively suppressing all cannibalism by 1956 [9]. One of the mostremarkable biological properties of kuru is its incredibly long incubation period. Living individuals with kuru wereidentified up to June of 2004 based on a 2006 study, even though they were born “before the cessation of cannibalismin the late 1950s”, and had probably acquired kuru around the time of cessation [10]. Amazingly, individuals withkuru are able to survive with this fatal disease for so long because the incubation period may be anywhere from 5 to50 years, though kuru “is uniformly fatal within 4-24 months of symptom onset” [9].

Ritualistic Endocannibalism

The way the Fore people practiced their endocannibalism caused high incidence of the disease among women andleft a society of motherless children to be raised by their father in many instances [11]. Even by 1982, long afterthe cessation of practices, it was unusual to see older women in the Fore tribe [11]. When it came to consumingthe dead, contrary to the assumption stated later in the Model section, not all dead were consumed within the Forepopulation, but “kuru victims were viewed favorably”, making their chance of consumption much higher than thosewho died from other factors [7]. The endocannibalism occurred as part of a mourning ritual of a deceased tribemember [8], and was seen as a form of respect for dead relatives [9]. When it came to the ritual, not all membersparticipated in the consumption.

“Rules for consumption were very specific. Body parts were allocated according to kinship rightsand gender. Small children were never given meat and were also kept away from the ceremony.Female relatives by marriage, who were the main consumers, could also request certain bodyparts.”[7]

The quote above illustrates how the dead were divided among tribe members. The primary consumers of the meatwere female relatives, female children above the age of 6, and male children around the age of 6-10 [10]. Around theage of 8-12, boys “moved to the men’s house...leaving behind the world of immaturity, femininity, and cannibalism”[7]. Because of the small age range in which males practiced cannibalism, the disease predominantly affectedwomen and children, with only 2% of cases being adult men [10]. Thus, the Fore population has a disproportionateamount of adult males when compared to adult females.

One factor that keeps the population a bit more proportionate is the Fore belief that kuru is a result of sorcery,and the implications that came along with the accusation of being a sorcerer [8]. “To cause kuru, a would-be-sorcererwould need to obtain a part of the victim’s body” and packed this in with leaves and made into a “‘kuru bundle”’,and place the bundle partially submerged into a swamp. The bundle was then shook daily until “the sympathetic

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kuru tremor was induced in his victim”, [8].As a result, relatives of the kuru victim tried to identify and kill the suspected sorcerer “if they could not bribe

or intimidate him to release a victim from the kuru spell” [8]. Various rituals of divination were used to identify asorcerer, one of which involved collecting water from various areas and giving it to the kuru victim. If one of thewater samples induced vomiting, it was thought that the sorcerer lived near the respective water source [8]. Killingof a sorcerer was called tukabu, and was a “ritualistic form of vendetta” [8]. The ritual was incredibly brutal, oftenincluding “crushing with stones the bones of the neck, arm, and thigh, as well as the loins, biting the trachea, andgrinding the genitalia with stones and clubs” [8]. Those who were found to be sorcerers were often adult men, andthe killing of the male sorcerers helped to maintain the sex ratio in the Fore population heavily impacted by the kurudeaths of their women.

ModelBased off of the standard S� I �R model, an initial model was constructed for the spread of kuru within the ForeTribe using S� I �D components as there is no recovery from kuru, only death. The model is as follows:

S = a(S+ I)�bS�dS(µDi)

I = dS(µDi)� (b + g)I

Di = (b + g)I �µDi

in which S(t) represents the number of susceptible people in the Fore tribe, or those who do not have kuru, I(t)represents the number of those infected in the Fore tribe, or those who do have kuru, and Di(t) represents the deadwho are infected with kuru, whether or not they died as a consequence of kuru or by natural causes. The constant ais the birth rate in the Fore tribe, which is related to the entire population, b is the natural death rate of the tribe, g isthe death rate by kuru, d is the infection rate of kuru, and µ is the consumption rate of the infected dead.However, the model was simplified to just the S� I components because this investigation is not focused around thechange of dead people in the Fore tribe, but rather the change of susceptible and infected individuals. In order toget rid of the Di equation and the Di terms in the other equations, it was assumed that consumption of the dead isimmediate. Therefore, Di = 0 which allows µDi to equal (b + g)I, and leads to the following simplified model:

S = a(S+ I)�bS�dS((b + g)I)

I = dS((b + g)I)� (b + g)I

The assumptions used in the creation of the final model are:

1. Only the female population is represented in the model because of the ritualistic complexities surroundingmales and endocannibalism of the Fore tribe, causing only a very small percentage of those infected with kuruto be males (a negligible amount)

2. No familial preference is taken into account

3. The dead are immediately consumed

4. Every dead person is consumed

5. All infected dead can transmit disease, whether kuru caused their death or otherwise

6. Infection does not affect fertility (there is no evidence of prion diseases affecting fertility)

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7. All newborns are born susceptible

8. Birth rate is higher than natural death rate, otherwise the population would always go to zero

Additionally, the empirical values used to plot the vector fields and other graphs in the Results section were arbitraryin nature, but chosen to be possible values if empirical data were actually attained.

ResultsIn order to determine the behavior of the model, the fixed points and nullclines of the model are found. The fixedpoints are determined to be

(S⇤, I⇤) = (0,0),(1d,

b �ad (a �b � g)

)

and the nullclines are found to beSnull : I =

bS�aSa �dbS�dgS

Inull : S =(b + g)I

d (b + g)I=

1d

Inull : I = 0

In order to determine the behavior at the fixed points, linear stability analysis will be utilized. The JacobianMatrix of the system is

J =h

a�b�db I�dgI a�dbS�dgSdb I+dgI dbS+dgS�b�g

i

Analysis of (0,0)At the fixed point (0,0), the Jacobian Matrix is

J(0,0) =h

a�b a0 �b�g

i

Therefore, the eigenvalues of this matrix are

l1 = a �b l2 =�b � g

Because a,b , and g are always positive, and because a > b , l1 is positive while l2 is negative. Therefore, asaddle node exists at the point (0,0), where l1 is unstable while l2 is stable. The eigenvectors associated with theseeigenvalues are

l1 :⇥

10⇤

l2 :⇥ �a

a+g⇤

Because our model is only interested in behavior in the first quartile and because the eigenvector associated with thestable eigenvalue is in the 2nd and 4th quartile, there will be no vectors pointing towards the fixed point in the firstquartile. Therefore, for the purposes of our model, this fixed point can be considered to be entirely unstable.

Analysis of ( 1d ,

b�ad (a�b�g) ) when a > b + g

When a > b + g , , b�ad (a�b�g) is negative while 1

d is positive. Therefore, this fixed point occurs in the 4th quartile, sothe behavior near the fixed point is not relevant to the behavior in the first quartile. Therefore, analysis of the vectorfield on the phase plane is used to determine the behavior of the model. Interestingly, the behavior of the modelwhen a > b + g is not trivial. As shown on the next page in figure 1, the vector field appears to converge and followa curve. Incidentally, this curve is actually the S nullcline. While none of these points are actually fixed points, themodel still appears to follow this curve, which results in a stabilization of the susceptible population over time whilethe infected population grows to infinity.

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(a) a = 2.5 b = 2 g = .4 d = .002 (b) a = 2.5 b = 2 g = .4 d = .002 with nullclines

Figure 1. On the left is a graph of the vector field with the assigned values shown in (a). Notice how these vectorsappear to converge on the S nullcline, as shown on the right.

Manipulation of gAs shown below in figure 2, altering the value for g results in a change in the behavior of the model. As g decreases,the S nullcline is significantly shifted to the right. Therefore, the susceptible population would stabilize at a largervalue over time as the infected population proceeds to infinity. In other words, as the death rate by kuru decreases,the number of susceptible members in the population increases.

(a) a = 2.5 b = 2 g = .4 d = .002 (b) a = 2.5 b = 2 g = .1 d = .002

Figure 2. Notice how a decrease in g causes a significant shift of the vector field and the nullcline to the right.

Manipulation of dAs shown on the next page in figure 3, altering the value for d results in a change in the behavior of the model. As dincreases, the S nullcline is significantly shifted to the left and down. Therefore, the susceptible population wouldstabilize at a smaller value over time as the infected population proceeds to infinity. In other words, as the rate ofinfection for kuru increases, the number of susceptible members in the population decreases.

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(a) a = 2.5 b = 2 g = .4 d = .002 (b) a = 2.5 b = 2 g = .4 d = .003

Figure 3. Notice how an increase in d causes a significant shift of the vector field to the left and down.

Plotting Solution Trajectories

In order to better visualize the behavior of the model when a > b + g , some solution trajectories given arbitraryinitial conditions are plotted below in figure 4. Notice how all of the trajectories, regardless of initial conditions,eventually follow the S nullcline.

550 600 650

1000

2000

3000

4000

(a) a = 2.5 b = 2 g = .4 d = .002

Figure 4. Notice how all trajectories follow the S nullcline

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Analysis of ( 1d ,

b�ad (a�b�g) ) when a < b + g

When a < b + g , the fixed point ( 1d ,

b�ad (a�b�g) ) is in the first quartile. Therefore, performing linear stability analysis

would be helpful in determining the behavior of the model. The Jacobian Matrix for the model at ( 1d ,

b�ad (a�b�g) ) is

J(1d,

b �ad (a �b � g)

) =

a�b� b (b�a)

a�b�g �g(b�a)a�b�g a�b�g

b (b�a)a�b�g

g(b�a)a�b�g 0

�

Therefore, the eigenvalues of the system are

l1,l2 =a + ag

a�b�g ±q(�a � ag

a�b�g )2 +8[(b + g)(b �a)]

2As shown below in figure 5, the determinant of the eigenvalues is negative assuming that a and b do not change

and that a < b + g . Additionally, the real portion of the eigenvalue is always negative under the same assumptions.Therefore, the fixed point at ( 1

d ,b�a

d (a�b�g) ) is always a stable spiral.

(a) The real portion of the eigenvalues with varying g (b) The determinant of the eigenvalues with varying g

Figure 5. Notice how the determinant of the eigenvalues is always negative and the real portion of the eigenvaluesis always negative. This indicates that the fixed point is always a stable spiral.

Manipulation of gAs shown on the next page in figures 6 and 7, altering the value for g results in a change in the behavior of the model.As shown by figure 5, as g increases, the I nullclines remains unchanged. However, the S nullcline is shifted over.Therefore, the fixed point moves downwards meaning that the new stable value has the same value for susceptiblemembers but a lower number of infected members. In other words, as the death by kuru rate increases, the populationapproaches a state where there are less infected individuals at any time. Additionally, as shown by figure 6, as gincreases, the spiral appears to become more circular rather than elliptical. In the context of the model, this wouldmean that the oscillations of the infected individuals would become less severe as the death by kuru rate increases,and the oscillations in the susceptible individuals would become more severe as the death by kuru rate increases.

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(a) a = 2.5 b = 2 g = .6 d = .002 (b) a = 2.5 b = 2 g = 1.2 d = .002

Figure 6. Notice how an increase in g causes a significant downward shift of the fixed point.

(a) a = 2.5 b = 2 g = .6 d = .002 (b) a = 2.5 b = 2 g = 1.2 d = .002

Figure 7. Notice how an increase in g causes the stable spiral to appear more rounded than elliptical

Manipulation of dAs shown on the next page in figure 8, altering the value for d results in a change in the behavior of the model. As dincreases, the vertical I nullcline shifts to the left and the S nullcline shifts to the left and slightly down. As a result,the fixed point has a lower value for both the susceptible population and the infected population. In other words,an increase in the infection rate of kuru leads to the population stabilizing at an altogether lower value, with thesusceptible population seeing the largest difference.

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(a) a = 2.5 b = 2 g = 1.2 d = .002 (b) a = 2.5 b = 2 g = 1.2 d = .003

Figure 8. Notice how an increase in d causes both nullclines to shift left, greatly reducing the fixed point value forthe susceptible population and also lowering the value for the infected population.

Plotting Solution Trajectories

In order to better visualize the behavior of the model when a < b + g , some solution trajectories given arbitraryinitial conditions are plotted below in figure 9. Notice how in the figure on the left, the solution trajectories spiralas they approach the fixed point, which further solidifies the evidence for the fixed point behaving as a stablespiral. Interestingly, the figure on the right shows that solution trajectories tend to follow a curve until they reach asufficiently close range of the fixed point. As it turns out, this curve is once again the S nullcline.

0 200 400 600 800 1000

200

400

600

800

1000

(a) a = 2.5 b = 2 g = 1.2 d = .002

200 400 500 600 700

0

500

1000

1500

2000

2500

(b) a = 2.5 b = 2 g = 1.2 d = .002

Figure 9. Notice how the figure on the left clearly displays the spiraling behavior of the fixed point. Also, thefigure on the right shows how trajectories follow a curve, the S nullcline, until the trajectories reach a certain rangeof the fixed point.

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DiscussionAccording to the model, both g and d prove to be highly influential in the behavior of the population. If a < b + g ,a stable spiral appears in the first quadrant, which is the biologically relevant portion of the model. Additionally, as gincreases, the S nullcline shifts such that the model approaches a lower population of infected individuals over time.Furthermore, as g increases, the vector fields take a more circular approach to the fixed point rather than an ellipticalapproach. As d increases, both nullclines shift such that the model approaches an overall lower population, with thesusceptible population experiencing the greatest change.

Biologically, the value for the death rate by kuru, g , is likely very small, due to the large incubation period ofkuru. Therefore, it is highly probable that the behavior of the population in reality is more similar to the behaviorof the model when the birth rate of the population is greater than the natural death rate plus the death rate by kuru(a > b + g). As a result, the number of susceptible members in the population is likely to remain relatively constant,as the total population continues to grow due to the growth in the infected population. Thus, the birth rate ofindividuals into the susceptible population and the rate in which individuals are becoming infected can be predictedto eventually reach an equilibrium.

While this model was designed to elucidate the disease spread of kuru, it would not be difficult to change thismodel to demonstrate the disease spread of other prion diseases, such as bovine spongiform encephalopathy, whichwas historically also spread by cannibalism [12]. Having knowledge on which parameters affect the long termbehavior of the disease spread provides invaluable insight on how to prevent or inhibit the spread of the disease.

The model is, of course, not perfect. The system only represented the female population, so a more accuratemodel would include the male population. Additionally, the model assumed endocannibalistic practices occurredinstantaneously as the infected individuals died. In reality, there is a lag time between the actual death of the infectedindividual and consumption of the individual. Finally, because the endocannibalistic practices are highly based onfamilial tradition, it would be greatly beneficial to expand the model by parsing the entire population into familialsubunits, and modeling the disease within and between those subunits in context of the entire population.

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References1. Kovacs, G. G. & Budka, H. Prion diseases: from protein to cell pathology. The Am. journal pathology 172,

555–565 (2008).

2. Gibbs Jr, C. J., Amyx, H. L., Bacote, A., Masters, C. L. & Gajdnsek, D. C. Oral transmission of kuru,creutzfeldt-jakob disease, and scrapie to nonhuman primates. J. Infect. Dis. 142, 205–208 (1980).

3. Prusiner, S. B. Prions. Proc. Natl. Acad. Sci. 95, 13363–13383 (1998).

4. Prusiner, S. B. et al. Molecular biology of prion diseases. Sci. 252, 1515–1522 (1991).

5. White, A. R. et al. Monoclonal antibodies inhibit prion replication and delay the development of prion disease.Nat. 422, 80–83 (2003).

6. Mathews, J., Glasse, R. & Lindenbaum, S. Kuru and cannibalism. The Lancet 292, 449–452 (1968).

7. Liberski, P. P. et al. Kuru: genes, cannibals and neuropathology. J. Neuropathol. & Exp. Neurol. 71, 92–103(2012).

8. Liberski, P. & Brown, P. Kuru: Its ramifications after fifty years. Exp. gerontology 44, 63–69 (2009).

9. Khan, Z. Z. Kuru. Background, Pathophysiol. Epidemiol. (2017).

10. Collinge, J. et al. Kuru in the 21st century—an acquired human prion disease with very long incubation periods.The Lancet 367, 2068–2074 (2006).

11. Prusiner, S. B., Gajdusek, D. C. & Alpers, M. P. Kuru with incubation periods exceeding two decades. Annalsneurology 12, 1–9 (1982).

12. Leiss, W. Mad cows and mother’s milk: the perils of poor risk communication (McGill-Queen’s Press-MQUP,2004).

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