Indirect Transmitted Infectious Diseases: from Microscopic Cycles to - PowerPoint PPT Presentation

Indirect Transmitted Infectious Diseases: from Microscopic Cycles to Macroscopic Cycles Jude D. Kong Department of Maths. and Stat. Sciences University of Alberta Indirect Transmitted Infectious Diseases November 12, 2013 1 / 24

John Snow’s Ghost Map Figure: Blue flags represent pumps for drinking water and red bars represent deaths at that address Indirect Transmitted Infectious Diseases November 12, 2013 2 / 24

Infectious diseases These are diseases caused by pathogenic micro organisms such as bacteria , virus, parasites and fungi; the disease can be spread directly or 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. Indirect Transmitted Infectious Diseases November 12, 2013 3 / 24

Our focus → bacteria � indirect transmission Reservior → bacteria � bacteriophage Bacteriophage → It is a virus that prey on bacteria. (b) bacterium (a) phage injecting it’s producing masses of DNA into a bacterium new viruses Indirect Transmitted Infectious Diseases November 12, 2013 4 / 24

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. Indirect Transmitted Infectious Diseases November 12, 2013 5 / 24

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 observed around the world (Emch, et al., 2008) Indirect Transmitted Infectious Diseases November 12, 2013 6 / 24

Human population No immunity Divide human population into two classes: susceptible and infected class Constant population Bacteria-phage system → predator-prey relationship Indirect Transmitted Infectious Diseases November 12, 2013 7 / 24

Block diagram µ Susceptible Infected α ( B ) popula- popula- tion(S) tion(I) φξ ξ γ Bacterial population(B) Phage population(P) δ logistic growth 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 Indirect Transmitted Infectious Diseases November 12, 2013 8 / 24

Incidence term � B < c ; 0 , α ( B ) = a ( B − c ) ( B − c )+ H , B ≥ c . K= pathogen carrying capacity ( 10 6 ) cell liter − 1 H= half saturation pathogen density, ( 10 6 − 10 8 ) cell litter − 1 a= maximum rate of infection, (0.1) day − 1 c= MID, ( 10 6 ) cell liter − 1 Indirect Transmitted Infectious Diseases November 12, 2013 9 / 24

Predation term BP γ K 1 + B γ = phage absorption rate (-) liter virion − 1 day − 1 β = phage burst size,(80-100) visions day − 1 K 1 = half saturation bacteria predation density, (-) cell (Jensen, et al., 2006) Indirect Transmitted Infectious Diseases November 12, 2013 10 / 24

System of equations dS dt = − α ( B ) S + µ I , dI dt = α ( B ) S − µ I , � � dB 1 − B B dt = rB K 1 + BP + ξ I , − γ K dP B dt = βγ K 1 + BP − δ P + φξ I N = S + I Indirect Transmitted Infectious Diseases November 12, 2013 11 / 24

Case 1a: No shedding with K ≤ c Only microscopic cycles are observed 8 5 6 x 10 x 10 x 10 5 10 100 7 4.5 6 4 9.9998 80 5 3.5 Susceptive population level Infected population level Bacteria population level Phage population level 3 9.9996 Susceptive 60 4 Infected Bateria 2.5 Phage 3 2 9.9994 40 1.5 2 1 9.9992 20 1 0.5 0 9.999 0 0 0 0 10 10 20 20 30 30 40 40 50 50 60 60 Time (days) Indirect Transmitted Infectious Diseases November 12, 2013 12 / 24

Case 1b: No shedding with K ≥ c B microscopic and macroscopic cycles are observed 8 5 5 6 x 10 x 10 x 10 x 10 6 10 5 8 7 5 9 4 6 Susceptive Susceptive population level 4 Infected Infected population level Bacteria population level Phage population level 5 Bateria 8 3 Phage 3 4 7 2 3 2 2 6 1 1 1 0 5 0 0 0 0 10 10 20 20 30 30 40 40 50 50 60 60 Time (days) Indirect Transmitted Infectious Diseases November 12, 2013 13 / 24

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 Indirect Transmitted Infectious Diseases November 12, 2013 14 / 24

Case 2a: Shedding with K ≤ c Only microscopic cycles are observed 7 5 5 x 10 x 10 x 10 4 10 100 14 3.5 12 9.9998 80 3 10 Susceptive population level Infected population level Bacteria population level Phage population level 2.5 9.9996 60 Susceptive 8 Infected Bateria 2 Phage 6 9.9994 40 1.5 4 1 9.9992 20 2 0.5 0 9.999 0 0 0 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 Time (days) Indirect Transmitted Infectious Diseases November 12, 2013 15 / 24

Case 2b: No shedding with K ≥ c Both microscopic and macroscopic cycles are observed 8 5 5 6 x 10 x 10 x 10 x 10 12 10.5 3 2.2 2 10 2.5 10 1.8 Susceptive 9.5 2 1.6 Infected Susceptive population level 8 Infected population level Bacteria population level Phage population level Bateria 1.4 Phage 9 1.5 6 1.2 8.5 1 1 4 0.8 8 0.5 0.6 2 7.5 0 0.4 0 7 −0.5 0.2 0 0 100 100 200 200 300 300 400 400 500 500 600 600 700 700 800 800 Time (days) Indirect Transmitted Infectious Diseases November 12, 2013 16 / 24

Observations Unlike the previous case, the period of these cycles if approximately 1 year, which corresponds to annual outbreaks observed 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 Indirect Transmitted Infectious Diseases November 12, 2013 17 / 24

chaos 5 x 10 2.5 2 1.5 Infected Population 1 0.5 0 −0.5 0 500 1000 1500 2000 2500 3000 Time (days) (a) shedding rate vs period with r=1 (b) trajectory for r=1 and ξ = 11 Indirect Transmitted Infectious Diseases November 12, 2013 18 / 24

5 x 10 3 2.5 2 Infected Population 1.5 1 0.5 0 0 500 1000 1500 2000 2500 3000 Time (days) (a) shedding rate vs period with r=5 (b) tracjectory for r=5 and ξ = 53 Indirect Transmitted Infectious Diseases November 12, 2013 19 / 24

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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 outbreaks in countries near the equator Indirect Transmitted Infectious Diseases November 12, 2013 21 / 24

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 Indirect Transmitted Infectious Diseases November 12, 2013 22 / 24

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 Indirect Transmitted Infectious Diseases November 12, 2013 23 / 24

Thank you for Llstening Indirect Transmitted Infectious Diseases November 12, 2013 24 / 24