Modeling Addiction with SIR Models and Di usion. David Zimmerman

Transcription

1 Modeling Addiction with SIR Models and Di usion David Zimmerman MTH 496: Senior Project Advisor: Dr. Shawn Ryan Spring

2 Abstract Contents 1 Introduction SIR Models Modeling Addiction Wangari-Stone Model Saturation Treatment Function Calculating Rate of Spread Qualitative Behavior Boundedness Fixed Points Bifurcations Bifurcations in Wangari-Stone Model Stability Modeling Spatial Interaction with Di usion Adding A Spatial Element to Our Model Reaction-Di usion Model Numerical Solution to Di usion Model Numerical Example Code Density Map Introduction 1.1 SIR Models A SIR Model is a compartmental ODE model for the spreading of disease. It compartmentalizes a population into three basic groups: S for those who are susceptible to a disease, I for those who are infected, R for those who are recovered, and N to represent the population. The transition rate from S to I is proportional to the product of the values of each. This is e ectively the law of mass action. 8 Ṡ = IS >< İ = IS I >: Ṙ = I It is assumed that each member of the population always transitions from S to I to R. Also, once you are recovered you cannot become susceptible again. 2

3 S I R The most basic SIR model is said to lack vital dynamics, meaning that births and deaths in the population are not considered. This is an example of a conservative system. Mathematically, we would say that the population N does not change in time and that dn = ds dt dt + di dt + dr dt = 0. We can easily modify this model to fit a di erent situation. What if, for example, once you are recovered you could become susceptible again? This situation assumes no immunity is obtained after infection and is modeled 8 Ṡ = IS + R >< İ = IS I. >: Ṙ = I R If we allow those who are recovered to become susceptible again our flow chart is cyclic. This modified version is often called an SIRS model. S R I We can also consider vital dynamics. If we enforce that the rate at which the population becomes susceptible (this could be the same the rate of birth) is equal to the mortality rate then we preserve the conservative nature of the system. 1.2 Modeling Addiction We can use a compartmental model of ODEs to describe the dynamics of substance addiction. It is reasonable to assume that the law of mass action still applies to the spread of addiction; someone must come into contact with an addict before they assume addiction. Addicts can die from overdosing, natural causese, or from other complications involving drug abuse and they can also recover. An interesting question would be if recovered addicts can become susceptible again or even more susceptible than they were before they became addicted. The model considered in the next section assumes that once you are recovered you cannot be susceptible again but can resume heroin use before the end of a treatment period. This situation is described with another modified SIR model where vital dynamics are considered. 3

4 1.3 Wangari-Stone Model The heroin epidemic model formulated by Wangari/Stone is considered here without alteration. The population is separated into four classes: S, susceptibles, U 1, heroin users not receiving treatment, U 2, heroin uses receiving treatment, and U 3, those successfully treated from heroin use. The following is assumed: 1. Individuals in all classes freely interact with one another 2. Addicts in treatment may or may not still be using heroin 3. Herion users in treatment relapse to heroin users not in treatment due to a decision to selfterminate treatment 4. Heroin users are separated from the population (e.g., treatment facility) and do not a ect susceptibles. Below is our model reflecting these assumptions: 8 Ṡ = U 1 S µs >< U 1 = U 1 S + U 2 (µ )U 1 T(U 1 ) U 2 = T(U 1 ) ( µ)u 2 >: U 3 = U 2 + U 1 µu 3 (1) The parameters and key quantities of system (1) are defined in the following table Parameter Description The rate at which individuals in a general population enter the system via the susceptible population (e.g., Births, coming of age) Contact rate between susceptibles and heroin users not receving treatment µ Death rate due to natural causes Rate at which those undergoing treatment relapse to heroin use Self-cure rate. Rate at which users recover without receiving treatment 1 Heroin related death rate of heroin users not in treatment 2 Heroin related death rate of heroin users in treatment Rate at which heroin users in treatment are successfully detoxified and cured from heroin use Rate at which heroin users are treated! The extent of saturation of heroin users within the community 4

5 Figure 1: Wangari-Stone Model 1.4 Saturation Treatment Function We have not yet defined the function T(U 1 ). This is the treatment function, or the rate at which users receive treatment. T(U 1 ) U 1 1 +!U 1 and we call this the saturation treatment function. This function is formulated by borrowing an idea from Michaelis-Menten kinetics. The rate that users are treated should depend on the availability of treatment and that rate should decrease as more resources are in use. When U 1 is small, treatment is approximately linear. T(U 1 ) U 1. When U 1 >> 1, T(U 1 )!, indicating that treatment centers are at capacity and the rate of treatment does not increase with the number of drug users. This feature is called saturation. 1.5 Calculating Rate of Spread Epidemiologists denote the number of secondary infections brought on by a single infective person by R 0 and call this the Basic Reproduction Number. In this context, we will interpret this as the expected value of secondary individuals infected by a single infected person. If this number is less than one, meaning you are unlikely to infect even one person, the disease (or addiction in our case) in the population is eradicated. Otherwise, the disease will spread. We can begin to calculate this number by observing that the average time an individual uses heroin is given by the formula 5

6 1 T 0 = µ The parameters µ, 1,, and all describe rates at which one exits the phase of untreated heroin use and have units 1. Hence this formula is the harmonic mean of these rates. Next, the probability time of surviving this period and receiving treatment is given by T 1 = T 0. The probability of surviving the heroin users in treatment phase and relapsing to heroin use is given by the rate at which heroin users stop receiving treatment (the harmonic mean of all the coe cients describing rates which members of the population cease receiving treatment) multiplied by the rate of relapse,. T 2 = µ Now, we can observe that total average time spent by users not in treatment on multiple passes is given by T = T 0 (1 + T 1 T 2 + (T 1 T 2 ) 2 + (T 1 T 2 ) ) which is a geometric series with ratio T 1 T 2 and T 1 T 2 < 1 since all parameters of (1) are nonnegative. This series is convergent and its sum can be calculated by the usual formula a where 1 r a = T 0. After some simplification this time can be written as ( µ) T = (µ )( µ) + ( µ)bf Next we multiply by the contact rate and average rate of recruitment /µ and we finally yield R 0 = 2 Qualitative Behavior ( µ) µ(µ )( µ) + µ ( µ). (2) Since (1) is a non-linear system we are restricted in the amount of quantitative information we can find and instead must focus on the qualitative behavior as parameters are varied. 2.1 Boundedness Proposition 2.1. The solutions of (1) are biologically feasible and are bounded from above and below: (a) Given S (0) > 0, U 1 (0) > 0, U 2 (0) > 0, U 3 (0) > 0, the trajectories (S (t), U 1 (t), U 2 (t), U 3 (t)) will remain positive for all time t. (b) If N(0) apple µ then N(t) apple µ for all time t where N = S + U 1 + U 2 + U 3. Proof. We will first prove (a). The solution to Equation 1 of (1) can be found by use of the integrating factor technique: S = S (0) exp h Z µt U 1 dt i + exp h Z µt U 1 dt i Z s exp h Z µs + U 1 ds i ds 0 6 0

7 U 2 and U 3 can be found in a similar way: 8 >< U 2 = U 2 (0)e ( µ)t + R t T(U 0 1(s))e ( µ)(s t) ds >: U 3 = U 3 (0)e µt + e R µt t ( U 0 2(s) + U 1 (s))ds Notice that if U 1 0, Then U 2 and U 3 will be greater than 0. If U 1 (0) > 0 and there exists a time t where U(t) < 0 then there must exist some time interval [0, t 0 ) where U 1 (t) > 0 and U 1 (t 0 ) = 0. If U 1 (t) < 0 for t > t 0 then U 1 (t 0 ) apple 0. However, U 1 (t 0 ) = U 2 (t 0 ) > 0, so t 0 does not exist. Thus, the trajectories are strictly positive for all time t. For (b) we observe that if we add all equations from (1) we get Ṅ = µn 1 U 1 2 U 2 < µn since 1, 2, U 1, and U 2 are non-negative. If we solve the inequality Ṅ < µn we get that N < N(0)e µt + (1 e µt ). It is clear that if N(0) < then N(t) < for all t. Moreover, if µ µ µ N(0) > then N(t) will enter the region (0, ) in finite time. µ µ 2.2 Fixed Points System (1) has fixed points that refer to addiction eradiction or sustained addiction over time. We obtain these points by setting the equations of (1) to 0, S? = U? 1 + µ, U? 2 = U? 1 ( µ)(1 +!U? (3) 1 ), U? 3 = U? 1 + ( µ)(1 +!U? 1 )U? 1 µ( µ)(1 +!U? 1 ). If we plug these in to equation 2 of (1) we get the expression for U? 1 where f (U? 1 ) = AU? BU? CU? 1 A = (µ )( µ)!, B = (µ )( µ) + µ!(µ )( µ) + ( µ) (4)!( µ), C = [(µ )( µ)µ + µ( µ)](1 R 0 ). U? 1 = 0 is always a fixed point which represents addiction eradication. To find other fixed points we can consider the equation 7

8 which has roots Notice that if! = 0, f (U? 1 ) = AU? BU? 1 + C = 0 (5) U? 1 1,2 = B ± p B 2 4AC. (6) 2A U? 1 = C (µ )( µ) + ( µ) which is only positive when R 0 > 1. For!>0, there are four interesting cases for U? 1 : (7) (i) B < 0 and (a) C = 0(R 0 = 1) or (b) B 2 4AC = 0. Both of these instances result in a root at U? 1 endemic equilibrium point. = 0 and a root at B 2A which is a unique positive (ii) C < 0(R 0 > 1) and B < 0. In this case p B 2 4AC > B, so there exists a biologically unfeasible negative equilibrium, a root at U? 1 = 0 and a positive endemic equilibrium. (iii) C > 0, B < 0, and B 2 4AC > 0. In this case p B 2 4AC < B, so there are two positive endemic equilbria. (iv) B > 0 and either (a) C > 0 or (b) B 2 < 4AC. This cases has no biologically feasible equilibrium points. 8

9 Figure 2: Cases (i)-(iv). Only the region to the right of the y-axis represents biologically feasible solutions to (1). 2.3 Bifurcations We can define a bifurcation of a dynamical system such as (1) as a smooth change in parameters that cause the system to suddenly change its qualitative or topological behavior. A simple example of this is the one dimensional system ẋ = x 2 + r Figure 3: The three behaviors of x 2 + r as r is varied This is an example of a saddle-node bifurcation. 9

10 Definition 1 (Saddle-Node Bifurcation). A saddle-node bifurcation occurs when, as a parameter is varied,two fixed points approach one another, collide, and mutually annihilate [6]. 2.4 Bifurcations in Wangari-Stone Model Our model has several bifurcation points as the composite parameters A, B, and C are varied. This results in the six di erent behaviors shown in Figure 2. It is straightforward to see that one of these points is when B < 0 and C = 0(R 0 = 1). As R 0 goes from being less than one to greater than one. The behavior of U 1 transitions from case (iii) to case (i.a) to case (ii). This is commonly called a backwards bifurcation and its behavior resembles that of a saddle-node bifurcation. It is preferable to think about this change in terms of the original model parameters. To do this for a system with multiple parameters we must choose a parameter to vary while holding others constant. Notice from (5) that when R 0 = 1 and when B < 0 (µ )( µ)µ + µ(µ ) = ( µ)!( µ) > (µ )( µ) + µ!(µ )( µ) + ( µ), which implies!> ( + µ!)(µ )( µ) + ( µ) ( µ) Together with the condition on C give us a critical value for!:.!> ( + µ!)(µ )( µ) + ( µ) µ ( µ) = (µ )( µ) + ( µ) µ ( µ) (8) :=! C This condition is necessary for the backwards bifurcation at R 0 = 1 to occur. Note that this parameter accounts for the extent of saturation of heroin users, so we can say that when this exceeds a certain threshold we will observe this bifurcation. 2.5 Stability Let s consider the heroin-free equilibrium (HFE). It is hard to compute and analyze the eigenvalues of the Jacobian of (1). It would make physical sense for the stability of the HFE to depend 10

11 on R 0. In particular it would make sense that this point would be stable for R 0 < 1 and unstable otherwise. However, R 0 is independent of our bifurcation parameter! so this is only the case when!<! c. If! 0 our saturated treatment function approximates a linear treatment function and heroin users are treated at a near constant rate making a endemic less likely. Theorem 2.1. The HFE is globally asymptotically stable i R 0 < 1 and!<! c, where R 0 is given by (2), and unstable otherwise. A rigorous proof is given in [1]. When!>! c it is not so straightforward. Case (iii) shows that two positive endemic equilibrium points can exist when R 0 < 1. Figure 4: R 0 = ,! = 0.3,! c = The above graph suggests stability of one of the endemic equilibrium points given in (6). The implication of this is that it is not enough to reduce R 0 below one to eradicate addiction in a population. Instead, R 0 must be reduced further to some value we denote R C 0. A full derivation of this is provided in [8]. 3 Modeling Spatial Interaction with Di usion 3.1 Adding A Spatial Element to Our Model So far our model has applied only to closed populations. Lets consider each population a subset of a larger population. Then it becomes natural to wonder how each population interacts with 11

12 another geographically given the freedom to cross spatial boundaries. Some questions we might ask ourselves are: 1. Do heroin users immigrate to regions where there are other heroin users? 2. Do those that are receiving treatment or those who are recovered move out of regions where there are high concentrations of heroin users? 3. If those in treatment move to a region where there is a low concentration of heroin users does it decrease the chance of relapse? 4. Instead of the above, are movements in and out of regions completely random? Does entropy play a role? We investigate at least a few of the questions in subsequent sections and how we might model these phenomena. 3.2 Reaction-Di usion Model It seems plausible that heroin users cluster together. Drug users tend to come from low income backgrounds which historically have higher concentrations of drug users. If a heroin user is among a low-income population it is unlikely for them to leave that population. If we make these assumptions then we can model this deterministic process by appending a term to (1) that satisfies a two-dimensional heat equation with Neumann boundary conditions. This requires that the region in which we solve the problem be a simply connected region in R 2 and our boundary to be piece-wise smooth. There is a subtlety here: If our region must be simply connected then how can we apply this model to places like Hawaii or Japan whose citizens freely travel across a network of islands? Or those who live in Manhattan and regularly commute to other areas in New York? It might possible to include travel routes as a part of our domain thus eliminating this problem. For now, we will ignore it. Lets consider only that regions with large concentrations of heroin users are attractors for heroin users in other regions. Our spatial model: 8 Ṡ = U 1 S µs >< U 1 = U 1 S + U 2 (µ )U 1 T(U 1 ) U 1 U 2 = T(U 1 ) ( µ)u 2 >: U 3 = U 2 + U 1 µu 3 where k, our di usion constant, is non-negative. 3.3 Numerical Solution to Di usion Model U 2 This section will explore how we might numerically solve (8). For the sake of simplicity a rectangular domain is chosen which contains all (and only) sub-populations of interest, is simply connected and piece-wise smooth. We will call this region. The strategy will be to solve (1) for 12 (9)

13 each sub-population using any suitable numerical method (Runge-Kutta, Forward-Euler, etc) for each sub-population, and then solve the di usion equation at each sub-population using forward Euler where the derivatives at each point will be approximated by finite di erence methods. We do not wish for members of any population to leave the domain, so we must enforce that the outward normal derivative at the boundary points is zero. Figure 5: Boundary Diagram A full system to solve the heat equation is given by: 1 = U 1 2 U 1 (x, y, 0) = A, an M N matrix containing [S (0), U 1 (0), U 2 (0), U 3 (0)] T ij >< for 0 apple i apple M, 0apple j apple 1 (0, y, t) 1(N, y, t) @U 1 (x, 0, t) >: 1(x, M, t) The di usion equation can be solved in the interior of A by discretizing the time domain and using the second order central di erence formula for the second derivative (10) U 1 U 1 k U 1(x x, y) 2U 1 (x, y) + U 1 (x + x, 2 ( x) 2 k U 1(x, y y) 2U 1 (x, y) + U 1 (x, y + y) ( y) 2 13

14 where x and y are the horizontal and vertical distances between sub-regions in and are represented by a single horizontal or vertical position in A. If we use forward Euler to solve (8) and denote system (1) by f the solution of the interior points in A is given by ~a ij, = ~a ij + tf(~a ij ) k t ( x) 2 (~a i 1, j 2~a i, j + ~a i+1, j ) k t ( y) 2 (~a i, j 1 2~a i, j + ~a i, j+1 ) (11) To guarantee that the solution will be stable over multiple time steps k t and k t must satisfy ( x) 2 ( y) 2 the Courant-Friedrichs-Lewy condition k t ( x i ) 2 < 1 2. Since the boundaries are defined piece-wise, we must enforce each boundary condition separately and we will have to be cautious with the corners of A since they lie on two boundaries. All the boundaries must satisfy the heat equation and have an outward derivative of zero. To do this we must use backwards/forwards di erence approximations using the values from the interior of A and set them equal to zero. The second order forward and backward di erence formulas are given by Forward Di erence: Backward Di 1 = 3U 1(x i ) 4U 1 (x i + x i ) + U 1 (x i + 2 x i i 2 x 1 = 3U 1(x i ) + 4U 1 (x i x i ) U 1 (x i 2 x i i 2 x i We can set these equations equal to zero and solve for U 1 (x i ) to compute the boundaries of A excluding the corners: Left Boundary: Right Boundary: Top Boundary: Bottom Boundary: ~a, i,1 = 4, ~a i,2 3 ~a, i,n = 4 ~a i,n 1 3 ~a 1, j = 4 ~a 2, j 3 ~a M, j = 4 ~a M 1, j ~a, i,3 ~a i,n 2 ~a 3, j 1 ~a M 2, j 3 Since the corners are on two boundaries, we must enforce that the directional derivatives are equal to zero. That is, 14

15 " # 1 Top Left Corner: ru 1 (0, 0) = 0 1 " 1 Top Right Corner: ru 1 (N, 0) = 0 1# " 1 Bottom Left Corner: ru 1 (0, M) = 0 1# " 1 Bottom Right Corner: ru 1 (N, M) = 0 1# The top right and bottom left corner can be computed using the forward and backward di erence methods previously described since each element in the direction vector has the same sign. The other two corners pose a problem as, when this method is employed, the point we are interested in end up canceling out. For the top left corner we can deal with this by 1 (0, 1( Similarly for the bottom right corner we 1 (N, 1(N Now we can compute the directional derivative by using a first order central di erence formula for the derivative of 1 = U 1(x i x i ) U 1 (x i + x i i 2 x i and a second order central di erence for the derivative of y. Now we can state the corners of A as Top Left Corner: Top Right Corner: Bottom Left Corner: Bottom Right Corner: ~a 1,1 = ~a 1,N ~a M,1 ~a M.N = x(4~a = 4( y~a 1,N 1 4( y~a M,2 = 2,1 ~a 3,1 x(4~a M 1,N 3 x y + x~a 2,N ) y~a 1,3 ) ( x~a 1,N 2 3( x + y) + x~a ( x~a M 1,1 ) 3( x + y) ~a M 2,N ) + 3 x + y M 2,1 + y~a M,N 2 + y~a 3,N ) y~a M,3 ) 3.4 Numerical Example Matplotlib has a built in function for turning matrices into color maps. This is a nice way to visualize how Equation 2 of (8) evolves in time. 15

16 4 Code Figure 6: Left: Colormap of initial conditions. Right: Forwad 10 units of time. All numerical computations and simulations where done using Python with the Numpy and Matplotlib packages. 4.1 Density Map import matplotlib.pyplot as plt import numpy as np Lambda=1.6; beta=0.001; delta_1=0.002; delta_2=0.001; mu=0.01; alpha=0.9;,! rhe=0.467; sigma=0.1; xi=0.015; omega=0.11 A = np.zeros([10,10,4]) for i in range(a.shape[0]): for j in range(a.shape[1]): A[i,j,0] = *np.random.normal(0,1) A[i,j,1] = *np.random.normal(0,1) A[i,j,2] = 15 + np.random.normal(0,1) A[i,j,3] = 10 + np.random.normal(0,1) def Heat_Map(A, dx, dy, D): def f(x_vec): S, U1, U2, U3 = x_vec return [Lambda - beta*u1*s - mu*s, beta*u1*s + rho*u2 - (mu + delta_1 + xi)*u1 - alpha*u1/(1 + omega*u1),,! alpha*u1/(1 + omega*u1) - M = A.shape[0] N = A.shape[1] (rho + sigma + delta_2 + mu)*u2, sigma*u2 + xi*u1 -,! mu*u3] 16

17 A_ = np.zeros([m, N, 4]) A_2d = np.zeros([m, N]) A_diff = np.zeros([m, N, 4]) time_steps = 100 time_length = 10 dt = time_length/time_steps rx = dt*d/dx**2 ry = dt*d/dy**2 for h in range(time_steps): for i in range(m): for j in range(n): eval = f(a[i,j]) for k in range(4): A_[i,j,k] = A[i,j,k] + dt*eval[k] for i in range(1,m-1): for j in range(1,n-1): A_diff[i,j,1] = A_[i,j,1] - rx*(a_[i-1,j,1],! -2*A_[i,j,1] + A_[i+1,j,1]) - ry*(a_[i,j-1,1],! -2*A_[i,j,1] + A_[i,j+1,1]) A_diff[i,j,0] = A_[i,j,0] A_diff[i,j,2] = A_[i,j,2] A_diff[i,j,3] = A_[i,j,3] #### BOUNDARY CONDITIONS #### for i in range(1,m-1): #Left column boundary condition A_diff[i,0,1] = (4/3)*A_diff[i,1,1] - (1/3)*A_diff[i,2,1] #Right column boundary condition A_diff[i,N-1,1] = (4/3)*A_diff[i,N-2,1] -,! (1/3)*A_diff[i,N-3,1] for j in range(1,n-1): #Top row boundary condition A_diff[0,j,1] = (4/3)*A_diff[1,j,1] - (1/3)*A_diff[2,j,1] #Bottom row boundary condition A_diff[M-1,j,1] = (4/3)*A_diff[M-2,j,1] -,! (1/3)*A_diff[M-3,j,1] #Corner boundary conditions for k in range(4): A_diff[0,0,k] = (dx*(4*a_diff[1,0,k] - A_diff[2,0,k]) -,! dy*a_diff[0,2,k])/(3*dx - dy) A_diff[0,N-1,k] = (4*(dy*A_diff[0,N-2,k] - dx*a_diff[1,n-1,k]),! - (dx*a_diff[0,n-3,k] + dy*a_diff[2,n-1,k]))/(3*(dx + dy)) A_diff[M-1,0,k] = (4*(dy*A_diff[M-1,1,k] - dx*a_diff[m-2,0,k]),! - (dx*a_diff[m-3,0,k] + dy*a_diff[m-1,2,k]))/(3*(dx + dy)) A_diff[M-1,N-1,k] = (dx*(4*a_diff[m-2,n-1,k] -,! A_diff[M-3,N-1,k]) + dy*a_diff[m-1,n-3,k])/(3*dx + dy) 17

18 A = A_diff for i in range(m): for j in range(n): A_2d[i,j] = A_diff[i,j,1] plt.imshow(a_2d, cmap = 'hot', interpolation = 'bilinear') plt.colorbar() Heat_Map(A, 1, 1, 0.01) 18

19 References [1] P. van den Driessche and James Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. In: Math. Biosci. 180 (2002). John A. Jacquez memorial volume, pp issn: doi: / S (02) url: [2] Leah Edelstein-Keshet. 4. An Introduction to Continuous Models. In: Mathematical Models in Biology. Chap. 4, pp [3] Leah Edelstein-Keshet. 6. Applications of Continuous Models to Population Dynamics. In: Mathematical Models in Biology. Chap. 6, pp [4] Leah Edelstein-Keshet. 9. An Introduction to Partial Di erential Equations and Di usion in Biological Settings. In: Mathematical Models in Biology. Chap. 9, pp [5] Maia Martcheva. 2. Introduction to Epidemic Modeling. In: An Introduction to Mathematical Epidemiology. Springer Publishing Company, Chap. 2. [6] Stephen Strogatz. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity). Westview Press, [7] Jinliang Wang et al. Qualitative and bifurcation analysis using an SIR model with a saturated treatment function. In: Math. Comput. Modelling (2012), pp issn: doi: /j.mcm url: [8] Isaac Mwangi Wangari and Lewi Stone. Analysis of a heroin epidemic model with saturated treatment function. In: J. Appl. Math. (2017), Art. ID , 21. issn: X. doi: /2017/ url: [9] Xu Zhang and Xianning Liu. Backward bifurcation of an epidemic model with saturated treatment function. In: J. Math. Anal. Appl (2008), pp issn: X. doi: /j.jmaa url: