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 () 2013.4.12 1 / 24

Outline 1. Introduction 1 2. Model 2 3. Dynamics of the model 3 4. Parameter estimation and simulation 4 5. Control and discussion 5 () 2013.4.12 2 / 24

1. Introduction Background: Patients, Schistosome, Snail () 2013.4.12 3 / 24

1. Introduction Background Schistosomiasis () 2013.4.12 4 / 24

1. Introduction Background Schistosomiasis One of the most prevalent parasitic diseases () 2013.4.12 4 / 24

1. Introduction Background Schistosomiasis One of the most prevalent parasitic diseases 207 million people () 2013.4.12 4 / 24

1. Introduction Background Schistosomiasis One of the most prevalent parasitic diseases 207 million people Difficult to eradicate () 2013.4.12 4 / 24

1. Introduction Background Schistosomiasis One of the most prevalent parasitic diseases 207 million people Difficult to eradicate Four factors in transmission: () 2013.4.12 4 / 24

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 () 2013.4.12 4 / 24

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 () 2013.4.12 4 / 24

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 () 2013.4.12 4 / 24

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 () 2013.4.12 4 / 24

1. Introduction Life cycle of schistosome () 2013.4.12 5 / 24

1. Introduction Two islets in Nanjing () 2013.4.12 6 / 24

1. Introduction Two islets in Nanjing Two islets: () 2013.4.12 7 / 24

1. Introduction Two islets in Nanjing Two islets: Qianzhou, Zimuzhou, in the center of the Yangtze River near Nanjing () 2013.4.12 7 / 24

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 () 2013.4.12 7 / 24

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 () 2013.4.12 7 / 24

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 () 2013.4.12 7 / 24

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 ? () 2013.4.12 7 / 24

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 ? () 2013.4.12 7 / 24

2. Model Model ✻ µ x x s A x ✻ µ x x i ✲ x s Rats α x x i β x x s y i ✲ x i ✲ α y y i β y x i y s A y θ y e ✛ ✛ ✛ ✛ y i y e y s Snails µ y y i µ y y e µ y y s ❄ ❄ ❄ Figure: The flow diagram of schistosomiasis activities. () 2013.4.12 8 / 24

2. Model Model dx s  dt = A x − µ x x s − β x x s y i ,      dx i   dt = β x x s y i − ( µ x + α x ) x i ,       dy s  (1) dt = A y − µ y y s − β y x i y s ,   dy e    dt = β y x i y s − ( µ y + θ ) y e ,      dy i   dt = θ y e − ( µ y + α y ) y i .   () 2013.4.12 9 / 24

2. Model Model dx s  dt = A x − µ x x s − β x x s y i ,      dx i   dt = β x x s y i − ( µ x + α x ) x i ,       dy s  (1) dt = A y − µ y y s − β y x i y s ,   dy e    dt = β y x i y s − ( µ y + θ ) y e ,      dy i   dt = θ y e − ( µ y + α y ) y i .   µ x , A y A x µ y : 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 ) () 2013.4.12 9 / 24

2. Model Parameters A x , per capita reproduction rate of rats; per capita natural death rate of rats; µ x , α x , per capita disease-induced death rate of rats; per capita contact transmission rate from infected snails to β x , susceptible rats; A y , per capita reproduction rate of snails; per capita natural death rate of snails; µ y , α y , per capita disease-induced death rate of snails; per capita contact transmission rate from infected rats to β y , susceptible snails; per capita transition rate from infected and preshedding snails θ, to shedding snails. () 2013.4.12 10 / 24

3. Dynamics of the model Existence of equilibria () 2013.4.12 11 / 24

3. Dynamics of the model Existence of equilibria The basic reproduction number: � A x A y θβ x β y R 0 = ρ ( FV − 1 ) = 3 µ x µ y ( µ x + α x )( µ y + α y )( µ y + θ ) . () 2013.4.12 11 / 24

3. Dynamics of the model Existence of equilibria The basic reproduction number: � A x A y θβ x β y R 0 = ρ ( FV − 1 ) = 3 µ x µ y ( µ x + α x )( µ y + α y )( µ y + θ ) . µ x , 0 , A y The disease free equilibrium E 0 = ( A x µ y , 0 , 0), The unique endemic equilibrium E ∗ = ( x ∗ s , x ∗ i , y ∗ s , y ∗ e , y ∗ i ) ⇐ R 0 > 1 () 2013.4.12 11 / 24

3. Dynamics of the model Stability of equilibria Using a Lyapunov function: s ln( x s i ln( x i = i { [ x s − x ∗ s − x ∗ s )] + [ x i − x ∗ i − x ∗ i )] } V β y y ∗ s x ∗ x ∗ x ∗ s ln( y s e ln( y e + β x x ∗ i { [ y s − y ∗ s − y ∗ s )] + [ y e − y ∗ e − y ∗ e )] s y ∗ y ∗ y ∗ + µ y + θ i ln( y i [ y i − y ∗ i − y ∗ i )] } , θ y ∗ and by LaSalle’s Invariance Principle, the stability is Theorem The disease free equilibrium E 0 of the system (1) is globally asymptotically stable if R 0 ≤ 1 . Theorem For system (1) , if R 0 > 1 , the endemic equilibrium E ∗ is globally asymptotically stable. () 2013.4.12 12 / 24

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) () 2013.4.12 13 / 24

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) () 2013.4.12 14 / 24

4. Parameter estimation and simulation Parameter estimation parameters values(per capita per day) references A x 0.00006 estimated; 9 . 13 × 10 − 4 Xugy,1999 µ x 8 . 33 × 10 − 5 α x Ishikawa,2006 0 . 007 estimated; β x A y 0 . 108 estimated; 2 . 63 × 10 − 3 Anderson,1992, Feng,2005 µ y 4 . 67 × 10 − 3 α y Feng,2005 and Zhou,1988 0 . 0009 estimated β y 2 . 5 × 10 − 2 Allen,2003 θ () 2013.4.12 15 / 24

4. Parameter estimation and simulation Simulation Figure: The trajectories of x i and y i 0.148 0.0325 0.146 0.032 0.144 0.0315 X[i](t) Y[i](t) 0.142 0.031 0.14 0.0305 0.138 0.03 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 t t () 2013.4.12 16 / 24

4. Parameter estimation and simulation Simulation Figure: The trajectories of x i and y i 0.148 0.0325 0.146 0.032 0.144 0.0315 X[i](t) Y[i](t) 0.142 0.031 0.14 0.0305 0.138 0.03 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 t t R 0 = 1 . 29 > 1, () 2013.4.12 16 / 24

4. Parameter estimation and simulation Simulation Figure: The trajectories of x i and y i 0.148 0.0325 0.146 0.032 0.144 0.0315 X[i](t) Y[i](t) 0.142 0.031 0.14 0.0305 0.138 0.03 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 t t R 0 = 1 . 29 > 1, Schistosomiasis will be prevalent on this two islets. () 2013.4.12 16 / 24

5. Control and discussion Control measures Rats: () 2013.4.12 17 / 24

5. Control and discussion Control measures Rats: Mousetraps () 2013.4.12 17 / 24