Nonstandard finite difference models of some differential equations in Life Sciences Jean M-S Lubuma Department of Mathematics and Applied Mathematics Faculty of Natural and Agricultural Sciences University of Pretoria (South Africa) 10th Blackwell-Tapia Conference Institute for Computational and Experimental Research in Mathematics (ICERM), Brown University, Providence, USA, 9-10 November 2018, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 1 / 34
My first contact with the NSFD method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 2 / 34
Motivation and outline Complexity of natural processes leading to challenging mathematical models; 1 Differential Equations cannot be completely solved by analytical techniques; 2 Fundamental importance of reliable numerical methods to gain insight 3 Aim: Overview of some NSFD schemes that are dynamically consistent : 4 One-dimensional first-order ODEs models: NSFD schemes are topologically dynamically consistent; One-dimensional second-order ODEs models: NSFD schemes preserve the principle of conservation of energy Ebola Virus Disease models: NSFD schemes replicate their dynamics; Reaction-diffusion & Cross-diffusion models: NSFD schemes preserve boundedness and positivity. Colony Collapse Desorder: NSFD schemes preserve the (fast) declines of honeybee colonies. More on NSFD schemes NSFD method in Africa 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 3 / 34
One-dimensional 1st-order examples: Decay and Logistic Exact scheme (Mickens, 1994) of the decay equation dy dt = − ay : (1) y k +1 − y k y k +1 − y k (1 − e − ah ) / a = − ay k or ( e ah − 1) / a = − ay k +1 (2) Exact scheme (Mickens, 1994) of the logistic equation dy dt = y (1 − y ) : (3) y k +1 − y k = y k (1 − y k +1 ) or y k +1 − y k = y k +1 (1 − y k ) (4) e h − 1 1 − e − h Nonstandard scheme: y k +1 − y k = y k (1 − y k +1 ) . (5) h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 4 / 34
One-dimensional 1st-order examples: M-M equation Exact scheme (Mickens, 2011) of the Michaelis-Menten (1913) (M-M) model du au dt = − 1 + bu : (6) ( b exp( − ah ) u k exp( bu k ) u k +1 − u k [ ] − bu k (1 − e − ah ) / a = W . (7) b (1 − e − ah ) / a NSFD schemes include (Chapwanya, JL and Mickens, 2012): u k +1 − u k ( e ah − 1) / a = − au k +1 1 + bu k ; (8) u k +1 − u k au k +1 ( e ah − 1) / a = − 1 + bu k +1 ; (9) u k +1 − u k au k (1 − e − ah ) / a = − 1 + bu k . (10) Exact schemes of various equations: see, for instance, Roeger (2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 5 / 34
One-dimensional 1st-order examples: dynamic consistency Each one of the previous NSFD scheme can be written as u k +1 = G ( h ; u k ) where the function G satisfies the conditions G (0; u ) = u and dG ( h ; u ) > 0 . (11) du This guarantees that the schemes: Are monotone dependent on initial values (Anguelov, JL, 2003): u 0 ≤ v 0 = ⇒ u k ≤ v k ⇒ u k Preserve monotonicity: u ( t ) increasing/decreasing = increasing/decreasing Are topologically dynamically consistent, thus elementary stable (Anguelov, JL and Shillor, 2011): Every map G ( h ) is topologically equivalent to each evolution operator S ( t ) i.e. µ ◦ S ( t ) = G ( h ) ◦ µ for some homeomorphism µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 6 / 34
One-dimensional 1st-order examples: Simulations D 1 1 0.5 0.5 0 0 y y −0.5 −0.5 −1 −1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 (a) (b) t t 1 2.5 2 1.5 0.5 1 0.5 y 0 y 0 −0.5 −1 −0.5 −1.5 −2 −1 −2.5 (c) 0 1 2 3 4 5 6 7 8 (d) 0 2 4 6 8 10 t t Figure: Decay equation : (a) Exact solution; (b) Forward Euler method; (c) NSFD with complex denominator; (d) Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 7 / 34
One-dimensional 1st-order examples: Simulations L 1.5 1.5 1 1 y y 0.5 0.5 0 0 −0.5 −0.5 (a) 0 1 2 3 4 5 6 7 8 (b) 0 1 2 3 4 5 6 7 8 t t 1.5 1 y 0.5 0 −0.5 0 1 2 3 4 5 6 7 8 (c) t Figure: Logistic equation: (a) Exact solution; (b) Forward Euler method; (c) NSFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 8 / 34
One-dimensional 1st-order examples: Simulations M-M 1 1 NSFD scheme, b = 1 NSFD scheme, b = 1 NSFD scheme, b = 0 NSFD scheme, b = 0 Exact scheme, b = 1 Exact scheme, b = 1 0.8 0.8 0.6 0.6 u u 0.4 0.4 0.2 0.2 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 t t 1 NSFD scheme, b = 1 NSFD scheme, b = 0 Exact scheme, b = 1 0.8 0.6 u 0.4 0.2 0 0 10 20 30 40 50 60 t Figure: M-M model: NSFD schemes (8)-(10) versus exact scheme (7). Thus M-M equation cannot be approximated by the decay equation: Michaelis-Menten (1913) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 9 / 34
NSFD Scheme: Quid? Mickens’ rules: Rule 1. The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives of the differential equation. Rule 2. Denominator functions for the discrete derivatives must, in gen- eral, be expressed in terms of more complicated functions of the step-sizes than those conventionally used. Rule 3. Nonlinear terms should, in general, be replaced by nonlocal dis- crete representations. Rule 4. Special conditions that hold for the solutions of the differential equations should also hold for the solutions of the difference scheme. Rule 5. The scheme should not introduce ghost or spurious solutions. (e.g. the second order Runge-Kutta method, 2( y k +1 − y k ) / h = f ( y k ) + f [ y k + hf ( y k )] , for the decay equation, has spurious fixed-points for h = 2 / a ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 10 / 34
NSFD Scheme: Quid? Definition (Anguelov and JL, 2001) A difference equation to determine approximate solutions y k to the solution 1 y ( t ) of a differential equation is called a nonstandard finite difference scheme if at least one of the following two conditions is met: The classical denominator h of the discrete derivative is replaced by a nonnegative function φ ( h ) such that φ ( h ) = h + O ( h 2 ) as h → 0 + . Nonlinear terms that occur in the right-hand side of the ODE are approximated in a nonlocal way, i.e., by a suitable function of several points of the mesh. A NSFD scheme is called qualitatively stable or dynamically consistent with 2 respect to some property P of the differential equation, whenever the discrete equation replicates the property P for every value of h . Remark 1: Our formal definition retains only Mickens’ Rules 2 and 3; Remark 2: Sufficent conditions for topological dynamic consistency in scalar case: u k +1 = G ( h ; u k ) , G (0; u ) = u and dG ( h ; u ) > 0 . (12) du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 11 / 34
One-dimensional 2nd-order models Exact scheme of the harmonic oscillator (Mickens, 1994) d 2 y dt 2 + ω 2 y = 0 : y k +1 − 2 y k + y k − 1 (4 /ω 2 sin 2 ( h ω/ 2) + ω 2 y k = 0 , or (conservation of energy: Anguelov, Kama, JL; 2005): ) 2 ) 2 ( y k +1 − y k ( y k − y k − 1 + ω 2 y k +1 y k = + ω 2 y k y k − 1 . 2 /ω sin( h ω/ 2) 2 /ω sin( h ω/ 2) NSFD scheme for the Karman-Guderley-type equation (Buckmire, 1994-2004) 1 d ( r dy ) − ω 2 y = 0 : r dr dr ( y k +1 − y k 1 y k − y k − 1 ) − ω 2 y k = 0 . log ( r k +1 / r k ) − r k log ( r k / r k − 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 12 / 34
Ebola Virus Disease: fast and slow transmissions JL, Tsanou et al. (2016): Can the consumption of contaminated bush meat, the funeral practices and the environmental contamination explain the recurrence and persistence of EVD outbreaks in Africa? π" infec*on" γ" S" I" R" µ" µ" µ+δ" ξ""(shedding)" σ" """"""α" P" D" (shedding )" η" b" Figure: (1) Transmission of deceased individuals and demographic process (Agusto, Gumel et al. (2015), Fasina et al. (2014), Ivorra et al. (2015)); (2) Infection through contaminated environment; (3) Provision of Ebola virus from consumption of bats, hunted meat, etc.; (4) Not included: compartments of vaccinated and trained individuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lubuma (UP) 2018 13 / 34
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