Discrete models of disease and competition R. Bravo de la Parra , M. - PowerPoint PPT Presentation

Discrete models of disease and competition R. Bravo de la Parra ⋆ , M. Marvá ⋆ , E. Sánchez † , L. Sanz † ⋆ Departamento de Física y Matemáticas, UAH, Alcalá de Henares, Spain † Departamento Matemática Aplicada, ETSI Industriales, UPM, Madrid, Spain logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 1 / 20

The university of Alcalá Opened up in 1499. Moved to Madrid city in 1836 (UCM). Back to Alcalá in 1970. logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 2 / 20

Empirical evidences of diseases/parasites-population interactions Plasmodium azurophilum Anolis gingivinus Anolis Watts Bemisia tabaci Tribolium confusum Adelina tribolii Wolbachia Tribolium castaneum Global change. logo.png Diseases as populations control. M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 3 / 20

Organization levels and time scales Hierarchical organization levels Different levels, different time scales Subgroups with strong interactions ECOSYSTEM Heterogeneity may define subgrups: Epidemiological state Individual traits COMMUNITY Spatial distribution Social status POPULATION INDIVIDUALS Objectives: Building up models that capture previous behavior. 1 logo.png Models linking levels. 2 M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 4 / 20

Discrete two time scales model Process Time unit Epidemics Comunity logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 5 / 20

Discrete two time scales model Process Time unit Epidemics Comunity t t + 1 Notation S : competition, demography F : epidemics Two time scales model or slow-fast system k times F ◦ F ◦ · · · ◦ F ( N t ) = S ◦ F ( k ) ( N t ) � �� � N t +1 = S ◦ logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 6 / 20

Dimension reduction process Given the prototype X t + 1 = S ◦ F ( k ) ( X t ) (1) (H1) ∀ X ∈ Ω N there exists ¯ F ( X ) := lim k →∞ F ( k ) ( X ) X t + 1 := S ◦ ¯ F ( X t ) auxiliary system (2) G E → Ω N such that ¯ (H2) If there exist Ω q ⊂ R q , where q < N and Ω N − → Ω q − F = E ◦ G Y t + 1 = G ◦ S ◦ E ( Y t ) reduced system , Y = G ( X ) slow variables (3) Theorem Assume H1, H2. Let Y ∗ ∈ R q be a hyperbolic equilibrium of (3), then X ∗ = E ( Y ∗ ) hyperbolic equilibrium of (2). 1 Under suitable convergence F ( k ) → ¯ F , for k large enough: k → X ∗ equil of (1). There exist X ∗ 1 The stability of Y ∗ , X ∗ , X ∗ k is the same. 2 logo.png The basins of attraction of X ∗ , X ∗ k cab described by that of Y ∗ . 3 M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 7 / 20

Single species two time scales epidemic model: basic dynamics Fast process : SIS epidemics  ( N, 0) , ifR 0 ≤ 1   S  F S − βIS  S + I + γI k →∞ F ( k ) ( S, I ) =   lim    ( ν, (1 − ν )) N, ifR 0 > 1 I + βIS   S + I − γI I  1 ν = R 0 Constant total population size: N = S ( t ) + I ( t ) 1 Where R 0 = β/γ is the basic reproduction number . 2 Slow process : Beverton-Holt population dynamics  0 if b ≤ 1      b N ( t + 1) = 1+ cN ( t ) N ( t ) , ⇒ lim t →∞ N ( t ) =   N ∗ = b − 1    if b > 1     c logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 8 / 20

F ( k ) N ( t ) � � Single species two time scales epidemic model: N ( t + 1 ) = S Complete system: we distinguish S and I b S  S ( t + 1 ) = 1 + c SS S ( t ) + c SI I ( t ) S ( t )     b I   I ( t + 1 ) = 1 + c IS S ( t ) + c II I ( t ) I ( t )   S ( t ) ≡ F S ( k ) ( S ( t ) , I ( t )) I ( t ) ≡ F ( k ) I ( S ( t ) , I ( t )) But also SIS is faster than population dynamics b S ν N ( t )  S ( t + 1 ) =   1 + c SS ν N ( t ) + c SI ( 1 − ν ) N ( t )     Auxiliary system: k → ∞   b I ( 1 − ν ) N ( t )   I ( t + 1 ) =   1 + c IS ν N ( t ) + c II ( 1 − ν ) N ( t ) b 1 N ( t ) b 2 N ( t ) Reduced system: N = S + I N ( t + 1 ) = 1 + c 1 N ( t ) + 1 + c 2 N ( t ) logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 9 / 20

Results 1: persistence or extinction If ν b S + ( 1 − ν ) b I ≤ 1 then t →∞ N ( t ) = 0 lim t →∞ ( S ( t ) , I ( t )) = ( 0 , 0 ) lim ⇔ For instance, if b S , b I < 1 b S > 1 and b I < 1 − ν b S 1 − ν If ν b S + ( 1 − ν ) b I > 1 then � (( b 2 − 1 ) c 1 + ( b 1 − 1 ) c 2 ) 2 + 4 ( b 1 + b 2 − 1 ) c 1 c 2 � � ( b 2 − 1 ) c 1 + ( b 1 − 1 ) c 2 + t →∞ N ( t ) = N ∗ = lim 2 c 1 c 2 t →∞ ( S ( t ) , I ( t )) ≈ ( ν N ∗ , ( 1 − ν ) N ∗ ) lim ⇔ For instance, if b S , b I > 1 b S < 1 and b I > 1 − ν b S logo.png 1 − ν M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 10 / 20

Results 2: disease mediated competition (affects growth) That is, c SS = c II = c SI = c IS , then N ∗ = b S − b I R 0 + b I − 1 1 e N ∗ b S − 1 b S − 1 df Disease reduced fecundity b I < b S Disease enhanced fecundity b I > b S logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 11 / 20

Results 2, disease mediated competition (affects growth) That is, c SS = c II = c SI = c IS and b I = α b S , then N ∗ = b S − α b S R 0 + α b S − 1 1 e N ∗ b S − 1 b S − 1 df logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 12 / 20

Two competing species: the Leslie-Gower model b 1 N 1 ( t + 1 ) = 1 + c 11 N 1 ( t )+ c 12 N 2 ( t ) N 1 ( t ) b 2 N 2 ( t + 1 ) = 1 + c 21 N 1 ( t )+ c 22 N 2 ( t ) N 2 ( t ) N 2 N 2 E ∗ 2 Case A Case B b i ≤ 1 then N i ( t ) → 0 Γ 1 Γ 2 b i > 1 species i can survive alone E ∗ 2 Γ 2 Γ 1 N 1 N 1 Forward bounded solutions E ∗ E ∗ 0 E ∗ 0 E ∗ 1 1 Solutions are eventually componentwise N 2 N 2 monotone E ∗ Case C 1 2 Case C 2 Γ 1 E ∗ Γ i points whose i -coordinate is held fixed Γ 2 Γ 1 2 Γ 2 by the map N 1 N 1 E ∗ E ∗ 0 E ∗ 0 E ∗ 1 1 J.M. Cushing et al, 2004. Some Discrete Competition Models and the Competitive Exclusion Principle JDEA, 10 (13-15): 1139-1151 HL Smith 1998. Planar competitive and cooperative difference equations. JDEA, 3(5-6):335-357 logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 13 / 20

Two time scales competition with specialist parasite N ( t + 1 ) = S ( F ( k ) ( N ( t ))) Species 1: S and I S ( t ) ≡ F ( k ) I ( t ) ≡ F ( k ) S ( S ( t ) , I ( t )) I ( S ( t ) , I ( t )) b S F ( k ) S ( S ( t ) , I ( t )) S 1 ( t + 1 ) = 1 + c SS F ( k ) S ( S ( t ) , I ( t )) + c SI F ( k ) I ( S ( t ) , I ( t )) + c S 2 N 2 ( t ) b I F ( k ) I ( S ( t ) , I ( t )) I 1 ( t + 1 ) = 1 + c IS F ( k ) S ( S ( t ) , I ( t )) + c II F ( k ) I ( S ( t ) , I ( t )) + c I 2 N 2 ( t ) b 2 N 2 ( t ) N 2 ( t + 1 ) = 1 + c 2 S F ( k ) S ( S ( t ) , I ( t )) + c 2 I F ( k ) I ( S ( t ) , I ( t )) + c 22 N 2 ( t ) logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 14 / 20

Reduced Discrete SIS-competition model ν b S N 1 ( t ) N 1 ( t + 1 ) = 1 + ( ν c SS + ( 1 − ν ) c SI ) N 1 ( t ) + c S 2 N 2 ( t ) ( 1 − ν ) b I N 1 ( t ) + 1 + ( ν c IS + ( 1 − ν ) c II ) N 1 ( t ) + c I 2 N 2 ( t ) b 2 N 2 ( t ) N 2 ( t + 1 ) = 1 + ( ν c 2 S + ( 1 − ν ) c 2 I ) N 1 ( t ) + c 22 N 2 ( t ) All solutions in R 2 + are Forward bounded 1 Eventually componentwise monotone 2 Besides If ν b S + ( 1 − ν ) b I ≤ 1 then N 1 ( t ) → 0 1 If b 2 ≤ 1 then N 2 ( t ) → 0 2 Otherwise, species can survive alone 3 R. Bravo de la Parra, M. Marvá, E. Sánchez, L. Sanz 2017. Discrete Models of Disease and Competition ( Discrete Dynamics in Nature and Society, Article ID 5310837 ) HL Smith 1998. Planar competitive and cooperative difference equations. JDEA, 3(5-6):335-357 logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 15 / 20

Reduced Discrete SIS-competition model N 2 N 2 E ∗ 2 Case A Case B Γ 1 Γ 2 E ∗ E ∗ E ∗ 2 3 3 Γ 2 Γ 1 N 1 N 1 E ∗ E ∗ E ∗ E ∗ 1 0 0 1 N 2 N 2 E ∗ 2 Case C 2 Case C 1 E ∗ 2 Γ 2 Γ 1 Γ 1 Γ 2 N 1 N 1 logo.png E ∗ E ∗ E ∗ E ∗ 0 1 0 1 M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 16 / 20

Reduced Discrete SIS-competition model N 2 E ∗ Case D E ∗ 2 3 E ∗ 4 N 1 E ∗ E ∗ 0 1 Figure: Case D . Bi-stability with interior equilibrium point logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 17 / 20

Disease modified competition (affects competitive abilities) b S = b I > 1 , c SS = 3 > c SI = 2 . 8 , c 2 S = 2 > c 2 I = 1 . 8 , b 2 = 5, c S 2 = c IS = c II = c I 2 = c 22 = 1, ν ∈ ( 0 , 1 ] 7 C 1 6 D A 5 B b S C 2 4 3 0 . 4 0 0 . 2 0 . 6 0 . 8 1 ν = 1 /R 0 Increasing R 0 improves the species 1 competition outcome. logo.png M. Marvá (U. de Alcalá) A discrete disease-competition model Osnabrück - April 2019 18 / 20