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Indirect Transmitted Infectious Diseases: from Microscopic Cycles to Macroscopic Cycles

Jude D. Kong

Department of Maths. and Stat. Sciences

University of Alberta

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John Snow’s Ghost Map

Figure: Blue flags represent pumps for drinking water and red bars represent deaths at that address

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Infectious diseases

These are diseases caused by pathogenic micro organisms such as bacteria, virus, parasites and fungi; the disease can be spread directly

• r indirectly from one person to another.

Direct contact transmission→ occurs when there is a physical contact between an infected person and a susceptible person. Indirect contact transmission→ occurs when there is no direct human-to-human contact. Contact occurs from a reservoir to contaminated surfaces or objects, or to vectors.

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Our focus → bacteria indirect transmission Reservior →bacteria bacteriophage Bacteriophage →It is a virus that prey on bacteria.

(a) phage injecting it’s DNA into a bacterium (b) bacterium producing masses of new viruses

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Microscopic cycles →b bacteriophage cycles Macroscopic cycles → human population cycles Minimum Infection Dose (MID)- The minimum amount of pathogen required to cause an infection in the host.

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Our objectives

Show that the cyclical outbreaks in endemic regions are driven by the cycles generated by the predator-prey relationship that exist between bacteriophage and bacteria. Demonstrate the importance of the relationship between the MID and the bacterial carrying capacity in relation to the existence of the cycles Attempt an explanation to the different nature of the outbreaks

• bserved around the world

(Emch, et al., 2008)Indirect Transmitted Infectious Diseases

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Human population

No immunity Divide human population into two classes: susceptible and infected class Constant population Bacteria-phage system → predator-prey relationship

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Block diagram

Susceptible popula- tion(S) Infected popula- tion(I) Bacterial population(B) Phage population(P) α(B) γ ξ φξ δ logistic growth µ δ =phage death rate, (0.5-7.9) visions day−1 ξ= pathogen shed rate (10-100) cell liter−1 day−1 µ = human recovery rate (0.1) day−1 φ=mean phage shed rate (10−6 − 1) visions cell−1

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Incidence term

α(B) =

• 0,

B < c;

a(B−c) (B−c)+H,

B ≥ c. K= pathogen carrying capacity (106) cell liter−1 H= half saturation pathogen density, (106 − 108) cell litter−1 a= maximum rate of infection, (0.1) day−1 c= MID, (106) cell liter−1

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Predation term

γ BP K1 + B γ = phage absorption rate (-) liter virion−1 day−1 β =phage burst size,(80-100) visions day−1 K1 = half saturation bacteria predation density, (-) cell (Jensen, et al., 2006)

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System of equations

dS dt = − α(B)S + µI, dI dt =α(B)S − µI, dB dt =rB

• 1 − B

K

• − γ

B K1 + BP + ξI, dP dt =βγ B K1 + BP − δP + φξI N =S + I

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Case 1a: No shedding with K ≤ c

Only microscopic cycles are observed

10 20 30 40 50 60 9.999 9.9992 9.9994 9.9996 9.9998 10 x 10

5

Susceptive population level 1 2 3 4 5 6 7 x 10

6

Bacteria population level 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

8

Phage population level 10 20 30 40 50 60 20 40 60 80 100 Infected population level Time (days) Susceptive Infected Bateria Phage

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Case 1b: No shedding with K ≥ c

B microscopic and macroscopic cycles are observed

10 20 30 40 50 60 5 6 7 8 9 10 x 10

5

Susceptive population level 1 2 3 4 5 6 7 8 x 10

6

Bacteria population level 1 2 3 4 5 6 x 10

8

Phage population level 10 20 30 40 50 60 1 2 3 4 5 x 10

5

Infected population level Time (days) Susceptive Infected Bateria Phage

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Observations

Microscopic cycles always exist and the infected class does not have much control over them The infected class is only affected when the bacterial level goes above the MID

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Case 2a: Shedding with K ≤ c

Only microscopic cycles are observed

10 20 30 40 50 60 70 80 9.999 9.9992 9.9994 9.9996 9.9998 10 x 10

5

Susceptive population level 2 4 6 8 10 12 14 x 10

5

Bacteria population level 0.5 1 1.5 2 2.5 3 3.5 4 x 10

7

Phage population level 10 20 30 40 50 60 70 80 20 40 60 80 100 Infected population level Time (days) Susceptive Infected Bateria Phage

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Case 2b: No shedding with K ≥ c

Both microscopic and macroscopic cycles are observed

100 200 300 400 500 600 700 800 7 7.5 8 8.5 9 9.5 10 10.5 x 10

5

Susceptive population level 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 x 10

6

Bacteria population level 2 4 6 8 10 12 x 10

8

Phage population level 100 200 300 400 500 600 700 800 −0.5 0.5 1 1.5 2 2.5 3 x 10

5

Infected population level Time (days) Susceptive Infected Bateria Phage

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Observations

Unlike the previous case, the period of these cycles if approximately 1 year, which corresponds to annual outbreaks

• bserved in some endemic areas

The cycles exist at low level and only enter human population when the bacterial levels increased passed the MID The bacterial population peak before the infected human population

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chaos

(a) shedding rate vs period with r=1

500 1000 1500 2000 2500 3000 −0.5 0.5 1 1.5 2 2.5 x 10

5

Time (days) Infected Population

(b) trajectory for r=1 and ξ = 11

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(a) shedding rate vs period with r=5

500 1000 1500 2000 2500 3000 0.5 1 1.5 2 2.5 3 x 10

5

Time (days) Infected Population

(b) tracjectory for r=5 and ξ = 53

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Conclusion

As the cycles in the bacteria-phage system exist in the in the absence of human contribution to the bacterial and phage levels , and because the bacterial cycles peak before the human cycles when they exist in both systems, we can conclude that it is the microscopic cycles that are driving the macroscopic cycles If the phage level could be enhanced in some way to keep the bacteria below c, then only microscopic cycles will always exist The existence of chaotic behaviour explains the lack of clear periodicity in some endemic areas and the unpredictable nature of

• utbreaks in countries near the equator

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Future work

Explicitly include the role of infection derived immunity through the use of a recovered class Determine the exact conditions for the existence of limit cycles Determine the relation between the amplitude and the period of the cycles to other parameters in the system Include alternate forms of phage predation

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Appreciation

I would like to express my sincere gratitude to my supervisor Dr. Hao Wang for his constant support, guidance and motivation Members of the Journal Club, University of Alberta

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Thank you for Llstening

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