Lyme Disease: A Mathematical Approach

Transcription

1 July 21, 2015

2 Overview Biology of Lyme Disease 1 Biology of Lyme Disease Borrelia burgdoferi Ixodes scapularis Hosts 2 3 Age-Structured Tick Class Model 4 Age-Structured Tick Class & Seasonality Model

3 Borrelia burgdoferi Ixodes scapularis Hosts Why is Lyme Disease Important to Study? The shortening of Winter in the North led to the following: Warmer temperatures have been predicted to both enhance transmission intensity and extend the distribution of diseases such a malaria and dengue as well. Climate change may open up previously uninhabitable territory for arthropod vectors as well as increase reproductive and biting rates, and shorten the pathogen incubation period.

4 Borrelia burgdoferi Ixodes scapularis Hosts Borrelia Burgdoferi

5 Borrelia burgdoferi Ixodes scapularis Hosts Ixodes scapularis Ixodes scapularis, the black-legged tick Can be found throughout the country including Texas Take a blood meal every time they molt Once infected, they are infected for life Must attach for 36 hours to transmit the bacteria No vertical transmission Questing season is changing due to climate change

6 Borrelia burgdoferi Ixodes scapularis Hosts Figure of Life cycle:

7 Borrelia burgdoferi Ixodes scapularis Hosts White footed Mouse, Peromyscus leucopus

8 Biology of Lyme Disease Borrelia burgdoferi Ixodes scapularis Hosts Unidentified Alternate Host

9 Borrelia burgdoferi Ixodes scapularis Hosts 1 Recent data shows that ticks quest at two distinct heights 2 This will help narrow down the search for the other hosts

10 for

11 di M = α β M (1 i M )i T µ M i M dt ) di T dt = β T (1 i T ) ((αi M ) + (1 α)i A µ T i T di A dt = (1 α) β A (1 i A )i T δ 1 µ A i A

12 with Seasonality di M = α β M (1 i M )i T µ M i M dt ) di T dt = β T (1 i T ) ((αi M ) + (1 α)i A µ T i T di A dt = δ 1(1 α) β A (1 i A )i T δ 1 µ A i A

13 middle 2 3 : δ 1 = { 1 if 61 t Otherwise

14 Parameters Parameter Definition ρ M Birth rate of the mice into the susceptible class ρ T Birth rate of the ticks into the susceptible class ρ A Birth rate of the alternate host into the susceptible class β M Contact Transmission Rate for the mice β T Contact Transmission Rate for the Tick β A Contact Transmission Rate for the alternate host α Proportion of the the ticks that have a preference for questing at lower heights µ M Death rate for the Mouse class µ T Death rate for the Tick class µ A Death rate for the Alternate Host class Table: Table of Variables & Parameters for

15 Basic Reproduction Number of a Infection The nondimensionalized system was reduced and rearranged into an equation for R 0, which determines whether or not there will be an epidemic. If R 0 < 1 then the disease will eventually die out of the population. If R 0 = 1 the disease remains at a constant level in the population. If R 0 > 1 the level of disease in the population will increase until there is an epidemic.

16 R 0 := α 2 β M β T µ A + (1 α) 2 β A β T µ M µ M µ T µ A

17 Scenarios in which the overall R 0 > 1 and an epidemic will occur in the community. for 10 years for 10 years 0.6 Infected Mouse 0.5 Infected Mouse Infected Tick Infected Tick 0.5 Infected Alternate 0.45 Infected Alternate Proportion of Total Population Proportion of Total Population Time (Days) Time (Days)

18 Overall R 0 > 1 and an epidemic occurs for 10 years for 10 years Infected Mouse Infected Tick Infected Mouse Infected Tick 0.3 Infected Alternate 0.25 Infected Alternate Proportion of Total Population Proportion of Total Population Time (Days) Time (Days)

19 Scenarios in which the overall R 0 > 1 and an epidemic will occur in the community Cont. for 10 years for 10 years Proportion of Total Population Infected Mouse Infected Tick Infected Alternate Proportion of Total Population Infected Mouse Infected Tick Infected Alternate Time (Days) Time (Days)

20 Overall R 0 < 1 and the disease dies out 0.1 for 10 years 0.1 for 10 years Infected Mouse 0.09 Infected Tick Infected Alternate 0.09 Proportion of Total Population Proportion of Total Population Infected Mouse Infected Tick Infected Alternate Time (Days) Time (Days)

21 Adding an invading species with R 0 > 1 can increase the level of infection for the initial host

22 Three Tick Stages

23 Modeling the Spread of Lyme Disease Diagram that incorporates Criss-Cross Infection

24 Modeling the Spread of Lyme Disease di M dt di TL dt di TN E dt = αβ M (1 i M )i TN µ M i M (1a) = β TL (1 i TL )i M η TL i TL µ TL i TL (1b) ) = β TN (1 i TN E i TN ) (αi M + (1 α)i A η TN i TN E µ TN i TN E (1c) di TN dt di TA = η TL i TL η TN i TN µ TN i TN (1d) = η TN (i TN + i TN E ) µ TA i TA (1e) dt ) di A dt = β A(1 i A ) (i TA + (1 α)i TN µ A i A (1f)

25 Parameters Parameter Definition ρ M Birth rate of the mice into the susceptible class ρ T Birth rate of the ticks into the susceptible class ρ A Birth rate of the alternate host into the susceptible Larvae class β M Contact Transmission Rate for the mice β TL Contact Transmission Rate for the larval tick β TN Contact Transmission Rate for the nymphal tick β A Contact Transmission Rate for the alternate host η TL Rate that the larvae molt into nymphs η TN Rate that the larvae nymphs into adults α Proportion of the the ticks that have a preference for questing at lower heights µ M Death rate for the Mouse class µ T Death rate for the Tick class µ A Death rate for the Alternate Host class Table: Table of Variables & Parameters for Model with 3 Tick Classes

26 Basic Reproduction Number of a Infection R 0 := { αβ M β L η TL max µ M (η TL + µ TL )(η TN + µ TN ), } β TN (1 α)β A η TN (η TN + µ TN )µ TA µ A

27 To illustrate that the disease will be endemic in the alternate host. We chose variables in the following manner in the following manner: higher β M lower β TL, β TN, and β A

28 Model with 3 Tick Classes α = 1, β M =.21, β TL =.00041, β TN =.00041, β A =.00041, R 0 = Model with 3 Tick Classes for 10 years 0.8 Proportion of Total Population Infected Mouse Infected larvae Exposed Nymph Infected Nypmh Infected Adult Infected Alternate Time (Days)

29 Model with 3 Tick Classes α = 1, β M =.01, β TL =.0041, β TN =.0041, β A =.0041, R 0 = Model with 3 Tick Classes for 10 years Proportion of Total Population Infected Mouse Infected larvae Exposed Nymph Infected Nypmh Infected Adult Infected Alternate Time (Days)

30 It s very beneficial for the mice to be able to sustain the disease, but not necessary. If the the disease is endemic to the mice population, then it s very likely that the disease will be endemic for the alternate host population.

31 Modeling the Spread of Lyme Disease di M dt di TL dt di TN E dt = δ 3 αβ M (1 i M )i TN µ M i M (2a) = δ 1 β TL (1 i TL )i M η TL i TL µ TL i TL (2b) ) = δ 3 β TN (1 i TN E i TN ) (αi M + (1 α)i A η TN i TN E µ TN i TN E (2c) di TN dt di TA = η TL i TL η TN i TN µ TN i TN (2d) = η TN (i TN + i TN E ) µ TA i TA (2e) dt ) di A dt = β A(1 i A ) (δ 5 i TA + δ 3 (1 α)i TN µ A i A (2f)

32 { 1 if active>182 active<283 δ 1 = 0 not active { 1 : active>119 and active<283 δ 3 = 0 not active { 1 active>274 active<346 oractive>41 active<161 δ 5 = 0 not active

33

34 Basic Reproduction Number of a Infection R 0 := 0 0 t 121 αβ M β TL η TL 121 t 274 (η TL + µ TL )(η TN + η TN )µ M { } αβ M β L η TL max µ M (η TL + µ TL )(η TN + µ TN ), β TN (1 α)β A η TN 274 t 283 (η TN + µ TN )µ TA µ A t t 365

35 Model with Seasonality: α = 1, β M =.0011, β TL =.0011, β TN =.0011, β A =.0011, R 0 =.8693 Proportion of Total Population Model with 3 Tick Classes and Seasonality for 10 years Infected Mouse Infected larvae Exposed Nymph Infected Nypmh Infected Adult Infected Alternate Time (Days)

36 Model with Seasonality: α =.5, β M =.011, β TL =.011, β TN =.011, β A =.011, R 0 =.7561 Proportion of Total Population Model with 3 Tick Classes and Seasonality for 10 years Infected Mouse Infected larvae Exposed Nymph Infected Nypmh Infected Adult Infected Alternate Time (Days)

37 Model with Seasonality: α = 0, β M =.0011, β TL =.0011, β TN =.0011, β A =.0011, R 0 =.7036 Proportion of Total Population Model with 3 Tick Classes and Seasonality for 10 years Infected Mouse Infected larvae Exposed Nymph Infected Nypmh Infected Adult Infected Alternate Time (Days)

38 Lyme disease is found in mice in Texas at low levels. The model indicates that it s very likely that there is a larger host that is also an effective carrier.

39 References Branda JA, Rosenberg ES Borrelia miyamotoi: A Lesson in Disease Discovery. Ann Intern Med. 159: Fukunaga M Genetic and Phenotypic Analysis of Borrelia miyamotoi sp. nov., Isolated from the Ixodid Tick Ixodes persulcatus, the Vector for Lyme Disease in Japan. Int. J. Syst. Bacteriol. 45: Rollend, L Transovarial transmission of Borrelia spirochetes by Ixodes scapularis: a summary of the literature and recent observations. Ticks Tick-borne Dis. 4(12):46-51.

40 References Cont. Anderson, J., L. Magnarelli Vertebrate host relationships and distribution of ixodid ticks (Acari: Ixodidae) in Connecticut, USA. Journal of Medical Entomology, 17: Bertrand, M., M. Wilson Microclimate-dependent survival of unfed adult Ixodes scapularis (Acari: Ixodidae) in nature: life cycle and study design implications. Journal of Medical Entomology. 33: