Modeling the schistosomiasis on the islets in Nanjing Longxing Qi - - PowerPoint PPT Presentation

Modeling the schistosomiasis on the islets in Nanjing

Longxing Qi

School of Mathematical Sciences, Anhui University LAMPS and Department of Mathematics and Statistics, York University

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Outline

1

• 1. Introduction

2

• 2. Model

3

• 3. Dynamics of the model

4

• 4. Parameter estimation and simulation

5

• 5. Control and discussion

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• 1. Introduction

Background: Patients, Schistosome, Snail

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• 1. Introduction

Background

Schistosomiasis

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• 1. Introduction

Background

Schistosomiasis

One of the most prevalent parasitic diseases

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• 1. Introduction

Background

Schistosomiasis

One of the most prevalent parasitic diseases 207 million people

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• 1. Introduction

Background

Schistosomiasis

One of the most prevalent parasitic diseases 207 million people Difficult to eradicate

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• 1. Introduction

Background

Schistosomiasis

One of the most prevalent parasitic diseases 207 million people Difficult to eradicate

Four factors in transmission:

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• 1. Introduction

Background

Schistosomiasis

One of the most prevalent parasitic diseases 207 million people Difficult to eradicate

Four factors in transmission:

Definitive hosts—-human, mammals

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• 1. Introduction

Background

Schistosomiasis

One of the most prevalent parasitic diseases 207 million people Difficult to eradicate

Four factors in transmission:

Definitive hosts—-human, mammals Intermediate hosts—-snails

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• 1. Introduction

Background

Schistosomiasis

One of the most prevalent parasitic diseases 207 million people Difficult to eradicate

Four factors in transmission:

Definitive hosts—-human, mammals Intermediate hosts—-snails Schistosome

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• 1. Introduction

Background

Schistosomiasis

One of the most prevalent parasitic diseases 207 million people Difficult to eradicate

Four factors in transmission:

Definitive hosts—-human, mammals Intermediate hosts—-snails Schistosome Water

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• 1. Introduction

Life cycle of schistosome

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• 1. Introduction

Two islets in Nanjing

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• 1. Introduction

Two islets in Nanjing

Two islets:

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• 1. Introduction

Two islets in Nanjing

Two islets:

Qianzhou, Zimuzhou, in the center of the Yangtze River near Nanjing

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• 1. Introduction

Two islets in Nanjing

Two islets:

Qianzhou, Zimuzhou, in the center of the Yangtze River near Nanjing No human residents or other livestocks

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• 1. Introduction

Two islets in Nanjing

Two islets:

Qianzhou, Zimuzhou, in the center of the Yangtze River near Nanjing No human residents or other livestocks Rattus norvegicus infected by schistosome

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• 1. Introduction

Two islets in Nanjing

Two islets:

Qianzhou, Zimuzhou, in the center of the Yangtze River near Nanjing No human residents or other livestocks Rattus norvegicus infected by schistosome Snails

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• 1. Introduction

Two islets in Nanjing

Two islets:

Qianzhou, Zimuzhou, in the center of the Yangtze River near Nanjing No human residents or other livestocks Rattus norvegicus infected by schistosome Snails

What will happen ?

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• 1. Introduction

Two islets in Nanjing

Two islets:

Qianzhou, Zimuzhou, in the center of the Yangtze River near Nanjing No human residents or other livestocks Rattus norvegicus infected by schistosome Snails

What will happen ? How to control schistosomiasis on this two islets ?

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• 2. Model

Model

Rats

✲

Ax xs

✻

µxxs

✲ xi ✻

µxxi

✲

αxxi Snails βxxsyi yi

❄

µyyi

✛

αyyi

✛

θye ye

❄

µyye

✛

ys βyxiys

❄

µyys

✛

Ay Figure: The flow diagram of schistosomiasis activities.

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• 2. Model

Model

                             dxs dt = Ax − µxxs − βxxsyi, dxi dt = βxxsyi − (µx + αx)xi, dys dt = Ay − µyys − βyxiys, dye dt = βyxiys − (µy + θ)ye, dyi dt = θye − (µy + αy)yi. (1)

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• 2. Model

Model

                             dxs dt = Ax − µxxs − βxxsyi, dxi dt = βxxsyi − (µx + αx)xi, dys dt = Ay − µyys − βyxiys, dye dt = βyxiys − (µy + θ)ye, dyi dt = θye − (µy + αy)yi. (1)

Ax µx , Ay µy : the density—-closely related to the area of the two islets.

Chunhua Shan and Professor Huaiping Zhu: The Dynamics of Growing Islets and Transmission of Schistosomiasis Japonica in the Yangtze River (To appear in Bulletin of Mathematical Biology)

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• 2. Model

Parameters

Ax, per capita reproduction rate of rats; µx, per capita natural death rate of rats; αx, per capita disease-induced death rate of rats; βx, per capita contact transmission rate from infected snails to susceptible rats; Ay, per capita reproduction rate of snails; µy, per capita natural death rate of snails; αy, per capita disease-induced death rate of snails; βy, per capita contact transmission rate from infected rats to susceptible snails; θ, per capita transition rate from infected and preshedding snails to shedding snails.

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• 3. Dynamics of the model

Existence of equilibria

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• 3. Dynamics of the model

Existence of equilibria

The basic reproduction number: R0 = ρ(FV −1) =

3

• AxAyθβxβy

µxµy(µx + αx)(µy + αy)(µy + θ) .

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• 3. Dynamics of the model

Existence of equilibria

The basic reproduction number: R0 = ρ(FV −1) =

3

• AxAyθβxβy

µxµy(µx + αx)(µy + αy)(µy + θ) . The disease free equilibrium E0 = ( Ax

µx , 0, Ay µy , 0, 0),

The unique endemic equilibrium E ∗ = (x∗

s , x∗ i , y∗ s , y∗ e , y∗ i ) ⇐R0 > 1

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• 3. Dynamics of the model

Stability of equilibria

Using a Lyapunov function: V = βyy∗

s x∗ i {[xs − x∗ s − x∗ s ln( xs x∗

s )] + [xi − x∗

i − x∗ i ln( xi x∗

i )]}

+βxx∗

s y∗ i {[ys − y∗ s − y∗ s ln( ys y∗

s )] + [ye − y∗

e − y∗ e ln( ye y∗

e )]

+ µy+θ

θ

[yi − y∗

i − y∗ i ln( yi y∗

i )]},

and by LaSalle’s Invariance Principle, the stability is Theorem The disease free equilibrium E0 of the system (1) is globally asymptotically stable if R0 ≤ 1. Theorem For system (1), if R0 > 1, the endemic equilibrium E ∗ is globally asymptotically stable.

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• 4. Parameter estimation and simulation

Data

The data are based on the investigation of Nanjing Institute of Parasitic Diseases in the period of 1996-1998.

Table: Dissection results of snails from Qianzhou and Zimuzhou islets in 1996-1998

Islet Year No.dissected No.positive (%) Qianzhou 1996 2677 53 (2.0) 1997 8205 53 (0.6) 1998 7538 234(3.1) Zimuzhou 1997 6324 25 (0.4) 1998 5440 27 (0.5)

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• 4. Parameter estimation and simulation

Data

Table: Dissection results of rats from Qianzhou and Zimuzhou islets in 1996-1998

Islet Year No.dissected No.positive (%) Qianzhou 1996.12-1997.3 69 43 (62.3) 1997.12-1998.3 53 34 (64.2) Zimuzhou 1997.12-1998.3 67 36 (53.7) Total 189 113 (59.8)

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• 4. Parameter estimation and simulation

Parameter estimation

parameters values(per capita per day) references Ax 0.00006 estimated; µx 9.13 × 10−4 Xugy,1999 αx 8.33 × 10−5 Ishikawa,2006 βx 0.007 estimated; Ay 0.108 estimated; µy 2.63 × 10−3 Anderson,1992, Feng,2005 αy 4.67 × 10−3 Feng,2005 and Zhou,1988 βy 0.0009 estimated θ 2.5 × 10−2 Allen,2003

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• 4. Parameter estimation and simulation

Simulation

Figure: The trajectories of xi and yi

0.03 0.0305 0.031 0.0315 0.032 0.0325 X[i](t) 2000 4000 6000 8000 10000 t 0.138 0.14 0.142 0.144 0.146 0.148 Y[i](t) 2000 4000 6000 8000 10000 t

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• 4. Parameter estimation and simulation

Simulation

Figure: The trajectories of xi and yi

0.03 0.0305 0.031 0.0315 0.032 0.0325 X[i](t) 2000 4000 6000 8000 10000 t 0.138 0.14 0.142 0.144 0.146 0.148 Y[i](t) 2000 4000 6000 8000 10000 t

R0 = 1.29 > 1,

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• 4. Parameter estimation and simulation

Simulation

Figure: The trajectories of xi and yi

0.03 0.0305 0.031 0.0315 0.032 0.0325 X[i](t) 2000 4000 6000 8000 10000 t 0.138 0.14 0.142 0.144 0.146 0.148 Y[i](t) 2000 4000 6000 8000 10000 t

R0 = 1.29 > 1, Schistosomiasis will be prevalent on this two islets.

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• 5. Control and discussion

Control measures

Rats:

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• 5. Control and discussion

Control measures

Rats:

Mousetraps

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day Expensive

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day Expensive

Snails:

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day Expensive

Snails:

Molluscicides

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day Expensive

Snails:

Molluscicides The death rate of snails: 0.8−1/day

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day Expensive

Snails:

Molluscicides The death rate of snails: 0.8−1/day Cheap

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day Expensive

Snails:

Molluscicides The death rate of snails: 0.8−1/day Cheap

Which one should we choose if only one of them can be chosen?

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day Expensive kx be the rate of control rats per day,

Snails:

Molluscicides The death rate of snails: 0.8−1/day Cheap

Which one should we choose if only one of them can be chosen?

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• 5. Control and discussion

Control measures

Rats:

Mousetraps The rate of capture rats: 0.16−0.35/day Expensive kx be the rate of control rats per day,

Snails:

Molluscicides The death rate of snails: 0.8−1/day Cheap ky be the rate of control snails per day.

Which one should we choose if only one of them can be chosen?

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• 5. Control and discussion

Control model

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• 5. Control and discussion

Control model

                             dxs dt = Ax − (µx + kx)xs − βxxsyi, dxi dt = βxxsyi − (µx + kx + αx)xi, dys dt = Ay − (µy + ky)ys − βyxiys, dye dt = βyxiys − (µy + ky + θ)ye, dyi dt = θye − (µy + ky + αy)yi. (2)

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• 5. Control and discussion

Control model

                             dxs dt = Ax − (µx + kx)xs − βxxsyi, dxi dt = βxxsyi − (µx + kx + αx)xi, dys dt = Ay − (µy + ky)ys − βyxiys, dye dt = βyxiys − (µy + ky + θ)ye, dyi dt = θye − (µy + ky + αy)yi. (2) The basic reproduction number for model (2): R∗

0 =

3

• AxAyθβxβy

(µx + kx)(µy + ky)(µx + kx + αx)(µy + ky + αy)(µy + ky + θ).

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• 5. Control and discussion

Sensitivity analysis

The flexibility of R∗

0 to kx and ky are given by:

ER∗ Ekx = −1 3kx( 1 µx + kx + 1 µx + kx + αx ), ER∗ Eky = −1 3ky( 1 µy + ky + 1 µy + ky + αy + 1 µy + ky + θ).

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• 5. Control and discussion

Sensitivity analysis

Figure: The plotting of | ER∗

Ek | over kx and ky

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• 5. Control and discussion

Sensitivity analysis

Figure: The plotting of | ER∗

Ek | over kx and ky

The blue curve: | ER∗

Ekx | over kx,

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• 5. Control and discussion

Sensitivity analysis

Figure: The plotting of | ER∗

Ek | over kx and ky

The blue curve: | ER∗

Ekx | over kx,

The red curve: | ER∗

Eky | over ky,

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• 5. Control and discussion

Sensitivity analysis

Figure: The plotting of | ER∗

Ek | over kx and ky

The blue curve: | ER∗

Ekx | over kx,

The red curve: | ER∗

Eky | over ky,

The intersection point: ky = 0.019.

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• 5. Control and discussion

Sensitivity analysis

R∗

0 is more sensitive to ky than to kx when ky > 0.019,

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• 5. Control and discussion

Sensitivity analysis

R∗

0 is more sensitive to ky than to kx when ky > 0.019,

To control snails is easier to eliminate schistosomiasis than to control rats as long as ky > 0.019.

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• 5. Control and discussion

Compare two control measures

Figure: The trajectories of xi, ye and yi

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• 5. Control and discussion

Compare two control measures

Figure: The trajectories of xi, ye and yi

The black curve: kx = 0.001 and ky = 0,

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• 5. Control and discussion

Compare two control measures

Figure: The trajectories of xi, ye and yi

The black curve: kx = 0.001 and ky = 0, The blue curve: kx = 0 and ky = 0.01< 0.019,

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• 5. Control and discussion

Compare two control measures

Figure: The trajectories of xi, ye and yi

The black curve: kx = 0.001 and ky = 0, The blue curve: kx = 0 and ky = 0.01< 0.019, The red curve: kx = 0 and ky = 0.036> 0.019.

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• 5. Control and discussion

Conclusion

If only one of control measures can be chosen

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• 5. Control and discussion

Conclusion

If only one of control measures can be chosen To control snails is more efficient than to control rats,

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• 5. Control and discussion

Conclusion

If only one of control measures can be chosen To control snails is more efficient than to control rats, Make sure the rate of controlling snails be greater than 0.019.

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• 5. Control and discussion

Thank you!

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