An Introduction to Mathematical Biomedicine Rolando Placeres Jiménez Departamento de Física, Universidade Federal de São Carlos, Brazil. Ceramic Laboratory, École Polytechnique Federal de Lausanne, Switzerland. rplaceres@df.ufscar.br

Discrete models Linear models. System of linear difference equations. Application: blood CO2 levels. Nonlinear difference equations. Steady states, stability and critical parameters. Period doubling and chaotic behavior.

Suppose a population of cell divide synchronously with each number producing ‘a’ daughter cells. Let us defined the number of cells in each generation with a subscript M1, M2,…, M2 are respectively the number of cells in the first, second,… nth generation. A simple equation relation successive generations is M n+1 =a M n (1) Let us suppose that initially there are M0 cells. How big will be the population after n generations? Applying equation (1) recursively results M n+1 = a n+1 M 0 (2) The magnitude of ‘a’ will determine whether the population grows or decrease with time. That is 𝑎 >1 M increases over successive generations, 𝑎 <1 M decreases, 𝑎=1 M is constant.

Example 1. An insect population We will consider that the parameters of this model are constant. an: number of females in the nth generation, f: number of progeny per female r: ratio of female to total adult population, m: fractional mortality of young individuals. a n+1 =(1−m)rf a n a n+1 = (1−m)rf n a 0

Example 2. Propagation of annual plants Plants produces seeds at the end of their growth season (say August), after which they die. A fraction of these seeds survives the winter, and some of these germinate at the beginning of the season (say May), giving rise to the new generation of plants. The fraction that germinates depends on the age of the seeds. Parameters 𝛾: number of seeds produced per plant in August, 𝛼: fraction of one-year-old seeds that germinate in May, 𝛽: fraction of two-year-old seeds that germinate in May, 𝜎: fraction of seeds that survive a given winter.

Variables p n : number of plants in generation n, δ n : number of one-year-old seeds in April. Equations p n+1 = ασγp n +βσ(1−α) δ n γ n+1 =σγ p n The same problem can be formulated as a second-order equation p n+1 = ασγp n +β 𝜎 2 (1−α)γ p n−1

Systems of Linear difference equations x n+1 = a 11 x n + a 12 y n y n+1 = a 21 x n + a 22 y n This system can be converted to a single higher-order equation: x n+2 = (a 11 + a 22 ) x n+1 + ( a 12 a 21 − a 11 a 21 ) y n We will look for the solution as x n =c λ n , then we obtain the characteristic equation: λ 2 − (a 11 + a 22 )λ+ ( a 11 a 21 −a 12 a 21 )=0 The solution of the characterisctic equation are called eigenvalues, and their properties will uniquely determine the behaviour of solution of the equation.

For such equations, the principle of linear superposition holds For such equations, the principle of linear superposition holds. We can conclude that the general solution is x n = A 1 λ 1 n + A 2 λ 2 n y n = B 1 λ 1 n + B 2 λ 2 n Real eigenvalues. For real values of λ the quantitative behaviour of a basic solution depends on whether λ falls into one of fourth possible ranges: For λ>1, λ n grows as n increases, that is x n =c λ n grows without bound. For 0<λ<1, λ n decreases to zero with increasing n. For -1<λ<0, λ n oscillates between positive and negative values while decleaning in magnitude to zero. For λ<−1, λ n oscillates but with increasing magnitude.

The cases λ=1, λ=0, λ=-1 correspond respectively to (1) static solution x=const, (2) x=0, and (3) an oscillation between x=c and x=−c . The dominant eigenvalue (the value of largest magnitude) has the strongest effect on the solution.

Complex eigenvalues: λ 1 =a+bi , and λ 2 =a−bi The solution is x n = B 1 r n cos nφ +i B 2 r n sin(nφ). The real and imaginary parts are thenselves solutions. Thus, the real value solution is a superposition of both: x n = C 1 r n cos nφ +i C 2 r n sin(nφ), where r= a 2 + b 2 and φ=tan(b/a). Complex eigenvalues are associaated with oscillating solutions. These solutions have growing or decreasing amplitude if r>1 and r<0 respectively; and constant amplitude if r=1. Only when and only when tan b a is a rational multiple of π and r=1, the solution is periodic.

Example 3. Red blood cell production In the circulatory system, the red blood cells are constantly being destroyed and replaced. Since these cells carry oxygen throughout the body, their number must be maintained at some fixe level. Assume that the spleen filter out and destroys a certain fraction of the cells daily and the bone marrow produces a number proportional to the number lost on the previous day. What would be the cell count on the nth day? Parameter f: fraction of red blood cells removed by spleen, r: production constant. Variables R n : number of red blood cells circulating on day n, M n : number of red blood cells produced by marrow on day n. The equations for R n and M n : R n+1 = 1−f R n + M n M n+1 =γf R n

Nonlinear difference Equations A nonlinear difference equation is any equation of the form x n+1 =f( x n , x n+1 , …) where x n is the value of x in generation 𝑛 where the recursion function 𝑓 depends on nonlinear combinations of its arguments. In relatively few case can an analytic solution be obtained directly when the difference equation is nonlinear. Thus we must generally be satisfied with determining something about the nature of solution or with exploring solution with the help of the computer.

Steady state In the context of difference equations, a steady state solution x is defined to be the value that satisfies the relation x =f( x ) and this frequently referred to as a fixed point of the function f. A steady state is termed stable if neighboring states are attracted to it and unstable if the converse is true. Let us assume that, given equation x =f( x ) we have already determined x , a steady state solution. Given some value of x n close to x , will x n tend toward or away from this steady state?

We start with a solution x n = x + x n where x n is a small quantity termed perturbation of the steady state x . x n+1 = x n+1 − x =f x n − x By exploiting the fact that x n is small quantity f x n =f x + x n ≈f x + df dx x +ϑ( x n 2 ) Thus we have that x n+1 ≈ df x dx x n We know that the solution of this linear equation will be decreasing whenever df x dx <1.

Example. Consider the equation x n+1 =r x n (1− x n ) and determine stability properties of its steady states. Solution x =r x (1− x ) x 1 =0, x 2 =1−1/r df dx =r−2r x x 1 is stable whenever r <1. x 1 is stable for 1<r<3. The simple map x n+1 =r x n 1− x n , x n ∈[0,1] It is known as the logistic map.

Bibliography L. Edelstein-Keshet. Mathematical Models in Biology (SIAM, New York, 2005). G. L. Baker and J. P. Gollub. Chaotic Dynamics: an introduction (Cambridge University Press, Cambridge, 1996).

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