Dynamical Modeling of Viral Spread

Transcription

1 Dynamics at the Horsetooth Volume 2, 21. Dynamical Modeling of Viral Spread Department of Mathematics Colorado State University Report submitted to Prof. P. Shipman for Math 54, Fall 21 Abstract. In this paper we study a three-component mathematical model for the spread of a viral disease in a population of spatially distributed hosts. The model is developed from the two-component model proposed by Tuckwell and Toubiana in 27. The positions of the hosts are randomly generated in a rectangular map. Within-host viralimmune system parameters are generated randomly to provide variability across the population. Encounters between any pair of individuals are evaluated according to a Poisson process. Viral transmission deps on the viral loads in donors and occurs with a given probability p trans. At any time, the values of the viral load (V ) and the immune system uninfected (T) and infected (T ) effectors for each individual are given by the solution of a system of three differential equations. We analyze the stability of the critical points P 1 and P 2 of the system and discuss numerical solutions for V, T and T obtained in Matlab. Keywords: Epidemic; Spatial stochastic model; Viral spread; Viral dynamics; Viral population dynamical model. 1 Introduction Mathematical models of viral dynamics are an important area of biomathematics. Such models can help in understanding the nature of infectious diseases and, as a result, in developing effective drug treatment. The immune system is a complex mechanism and to model its response to viral infection in every detail is a too complicated task. Tuckwell and Toubiana (27) proposed to consider spatial locations of the hosts combined with statistical distributions of the dynamics parameters to provide variety in the population immune properties. They described a mathematical model for a simplified two-dimensional system of effectors and virus contained in each individual and mentioned that it can be easily incorporated for a three-components system. So, in this paper we consider a model of each host in the population at time t containing virus concentration V (t), uninfected but susceptible effectors (T-cells) T(t) and productively infected effectors (T-cells) T (t). Using the notation of Stafford et al. (2), in the absence of interaction between hosts, V (t), T(t) and T (t) evolve according to the equations 1

2 dt = λ dt ktv, dt (1) dt = ktv δt, dt (2) dv dt = πt cv ktv, (3) where λ is the rate of production of effectors, d is the per capita removal (death) rate of effectors, k determines the rate of production of effectors per unit amount of virus. Productively infected cells produce virions at the rate π and die with rate δ per cell, virus is cleared with rate constant c. The system has two critical points P 1 = ( λ d,, ) and P cδ 2 = ((π δ)k ), λ δ dc (π δ)k, λ(π δ) cδ d k ). If λ =, P 1 is at the origin and is an asymptotically stable node. Otherwise, it is a saddle point if 1 < (π δ)kλ dδc, and the disease is promoted. If the latter inequality reversed, P 1 is an asymptotically stable node, and the disease is demoted. Note that since the associated eigenvalues for the system (1)-(3) are always real, P 1 can not be a spiral point. There are three possibilities for the second critical point P 2 : unstable saddle, stable node or stable spiral point. We note that P 2 may occur at unphysical values of T and V. The condition for P 2 to occur at physical values, namely 1 < (π δ)kλ dδc, is exactly the condition for P 1 to be an unstable saddle point. For a detailed analysis of the nature of equilibria, see Tuckwell and Wan (2). We will see in the following section that for the parameters generated from the 1-patient data given by Stafford et al. (2), P 1 is a saddle point and P 2 is a stable spiral point more than 9% of the time. 2 The Mathematical Model Y X Figure 1: Random spatial host population of n = 1 individuals with coordinates (X i, Y i ), i = 1,...,n. A red circle marks a randomly chosen initially infected host. Consider a two-dimensional habitat, where locations of n individuals are determined by the coordinates (X i, Y i ), i = 1,...,n. X i and Y i are taken to be uniformly distributed on (, a) and Dynamics at the Horsetooth 2 Vol. 2, 21

3 Parameter Mean Standard Min Max deviation d, day λ, d (1 cells µl ) (µl k,( virions day ) x δ, day π, virions x day c, day Table 1: Distibutions of viral and host immune system parameters. (, b), respectively. A typical random spatial distribution for n = 1 and a = b = 1 can be seen in Figure 1. When there is no interaction between hosts, the stochastic differential equations describing the evolution of the viral and effectors population in the ith individual of the habitat are given by: dt i dt = λ i d i T i k i T i V i, (4) dt i and has equilibria at P 1,i = ( λ i d i,, ) and P 2,i = ((π i δ i )k i ), λ i = k i T i V i δ i T i, (5) dt dv dt = πt i c i V i k i T i V i, (6) c i δ i δ i d ic i (π i δ i )k i, λ i(π i δ i ) c i δ i d i k i ). To provide variability across the habitat, the nonegative parameters λ i, d i, k i, π i, δ i, c i are randomly distributed. For the purpose of this study, we used the parameters obtained from the 1-patient data given by Stafford et al. (2), see Table 1. Figures 2 and 3 show examples of randomly distributed positions of the critical points P 1,i and P 2,i, respectively, in the (T, T, V )-space for n = 1. Note that P 2,i can have unphysical (negative) values. Recall that there are two possibilities for P 1 : unstable saddle point or stable node whereas for P 2 there are three posibilities: unstable saddle, stable node or stable spiral point. In Figures 4 and 5 we show some examples of distribution of types of critical points. In the given examples, P 1 is a saddle point in 99% cases and P 2 is a stable spiral point in 93% cases. We assume that the encounters between individuals is governed by the Poisson process {N ij (t), t > }, i, j = 1,...,n with the rate parameter λ ij = Λ exp[ αd ij ], where d ij = (Xi X j ) 2 + (Y i Y j ) 2 is the distance between ith and jth hosts. Λ = 2 is the basic rate of meetings per day, α = ln1 is the decay of contact rate in space. So, in the presence of interaction between hosts, we rewrite equations (7)-(9) in the following form: dt i dt = λ i d i T i k i T i V i, (7) dt i = k i T i V i δ i Ti, (8) dt dv n dt = πt i c i V i k i T i V i + T ji. (9) j=1 Dynamics at the Horsetooth 3 Vol. 2, 21

4 Here T ji = βh(u(, 1) p trans )H(V j V crit ) determines the transmission from the jth to the ith individual if a meeting occurs between them. The standard values of the parameters involved: transmitted viral load β = 1, threshold viral load for transmission V crit = 3, probability of transmission of virus on contact p trans varies from.5 to.7, and U(, 1) is a random variable uniformly distributed on (, 1). The form of H(x y) is chosen to be a step function: { 1, if x y H(x y) =, otherwise. Note that T ii = for all i. To start, we assume all T i () = 1 and Ti () =. We choose randomly just one infected individual with a set of random dynamical parameters. Therefore, we have all V i () = except for some j i with < V j < V init (usually V init = 3). Suppose the time interval of viral spread is (, T max ] and time step is t (usually t = 1 day.) At each time step, we define nxn matrix M such that { 1, if individual i meets individual j, M ij =, otherwise. So, the value of M ij = 1 if a uniform on (, 1) random number is less than λ ij t. Note that M is symmetric. At each time step, we update the ith individual s V i, T i, and Ti values according to the following: { T i (t), if individual i has never been infected, T i (t + t) = T i (t) + (λ i d i T i (t) k i T i (t)v i (t)) t, otherwise, { Ti (t + t) = Ti (t), if individual i has never been infected, V i (t + t) = T i (t) + (k it i V i (t) δ i T i (t)) t, { V i (t) + (πt otherwise, if n j=1 M ji =, i (T) c iv i (T) k i T i (T)V i (T)) t, V i (t) + (πti (T) c iv i (T) k i T i (T)V i (T)) t + n j=1 T ji(t), otherwise. 3 Results We did some numerical experiments varying time in days T max and dynamical parameters, see examples in Figures 6-9 for T max = 4 and in Figures 1-11 for T max = 1. For the future work, it is interesting to consider different values of probability of transmission of virus on contact p trans and population sizes. Dynamics at the Horsetooth 4 Vol. 2, 21

5 1.5 V T* T 3 4 Figure 2: Positions of the critical poits P 1,i according to the random distribution of the paramaters described in Table 1, i = 1,..., V T* T Figure 3: Positions of the critical poits P 2,i according to the random distribution of the paramaters described in Table 1, i = 1,...,1. Dynamics at the Horsetooth 5 Vol. 2, 21

6 1.2 SADDLE STABLE NODE 1.8 Frequency Type of Critical Point Figure 4: The numbers of each type of critical point for P SADDLE STABLE NODE STABLE FOCUS.8 Frequency Type of Critical Point Figure 5: The numbers of each type of critical point for P 2. Dynamics at the Horsetooth 6 Vol. 2, 21

7 Mean Population Viral Load Time in Days Figure 6: Mean population viral load vs. time for population with size n = 1 and p trans = Initially infected individual Individual infected later 4 35 Individual Viral Loads Time in Days Figure 7: Viral loads of two individuals vs. time Number of Sick INdividuals Time in Days Figure 8: Plots of numbers of individuals classified as sick with V > V sick vs. time, V sick = 5, p trans =.2. Dynamics at the Horsetooth 7 Vol. 2, 21

8 Mean Population Viral Load Time in Days Figure 9: Mean population viral load vs. time for population with size n = 1 and p trans = Initially infected individual Individual infected later 2 Individual Viral Loads Time in Days Figure 1: Viral loads of two individuals vs. time Number of Sick INdividuals Time in Days Figure 11: Plots of numbers of individuals classified as sick with V > V sick vs. time, V sick = 5, p trans =.2. Dynamics at the Horsetooth 8 Vol. 2, 21

9 Appix: Matlab Code %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Routine project.m solves system of n x 3 d i f f e r e n t i a l equations % which describe v i r a l spread in the popultion of density n clear ; % Random population map n = 1; X = rand (n, 2 ) ; figure ; hold on ; for i = 1:n plot (X( i,1),x( i,2), bx ) ; axis ([ 1 1 ] ) ; xlabel ( X ) ; ylabel ( Y ) ; hold o f f ; indinf = randi (n ) ; % index of the infected individual % Plotting the population map figure ( ) ; hold on ; for i =1:1 i f i = indinf plot (X( i,1),x( i,2), kx ) ; e l s e plot (X( indinf,1),x( indinf,2), ro ) ; axis ([ 1 1 ] ) ; xlabel ( X ) ; ylabel ( Y ) ; hold o f f ; % I n i n t i a l values and parameters [ d, lambda, k, delta, Pi, c ] = parameters (n ) ; tmax = 4; % time in days T = ones (n, tmax ) ; TI = zeros (n, tmax ) ; V = zeros (n, tmax ) ; V( indinf,1) = 3 rand ; % infected individual i n i t i a l v i r a l load L = 2; alpha = log (1); beta = 1; Vcrit = 3; Dynamics at the Horsetooth 9 Vol. 2, 21

10 Vsick = 5; ptrans =. 2 ; for t =1:tmax 1 % Generating meeting matrix M for i = 1:n 1 for j = i +1:n d i f f = X( i, : ) X( j, : ) ; % difference dd( i, j ) = sqrt ( d i f f diff ) ; % distance dd( j, i ) = dd( i, j ) ; l ( i, j ) = L exp( alpha dd( i, j ) ) ; l ( j, i ) = l ( i, j ) ; num = rand ; i f num< l ( i, j ) M( i, j ) = 1; e l s e M( i, j ) = ; M( j, i ) = M( i, j ) ; for i =1:n i f V( i, t)>1ˆ 8 T( i, t+1) = T( i, t)+lambda( i ) d( i ) T( i, t) k( i ) V( i, t ) T( i, t ) ; TI( i, t+1) = TI( i, t)+k( i ) V( i, t ) T( i, t) delta ( i ) TI( i, t ) ; e l s e T( i, t+1) = T( i, t ) ; TI( i, t+1) = TI( i, t ) ; summ = sum(m( i, : ) ) ; i f summ == V( i, t+1) = V( i, t ) + Pi ( i ) TI( i, t) c ( i ) V( i, t) k( i ) T( i, t ) V( i, t ) ; e l s e VT = ; for m=1:n i f V(m, t ) >= Vcrit rand num = rand ; i f rand num >= ptrans VT = VT + beta ; V( i, t+1) = V( i, t ) + Pi ( i ) TI( i, t) c ( i ) V( i, t) k( i ) T( i, t ) V( i, t)+vt; i f V( i, t+1)< V( i, t+1)=; Dynamics at the Horsetooth 1 Vol. 2, 21

11 for i =1:tmax ViralLoad ( i ) = mean(v( :, i ) ) ; % Plotting Mean Population Viral Load figure ( ) ; hold on ; plot ( 1 : 1 : tmax, ViralLoad, b ) xlabel ( Time in Days ) ; ylabel ( Mean Population Viral Load ) ; hold o f f ; % PLotting v i r a l loads of the i n i t i a l l y infected and a randomly chosen % individuals index=randi (n ) ; figure ( ) ; hold on ; plot ( 1 : 1 : tmax,v( indinf, : ), b.,1:1: tmax,v( index, : ), m ) xlabel ( Time in Days ) ; ylabel ( Individual Viral Loads ) ; leg ( I n i t i a l l y infected individual, Individual infected later, 1 ) ; hold o f f ; % Plotting number of sick individuals sumsick = [ ] ; for i =1:tmax indsick = find (V( :, i )>Vsick ) ; sumsick = [ sumsick length ( indsick ) ] ; figure ( ) ; hold on ; plot ( 1 : 1 : tmax, sumsick, g ) xlabel ( Time in Days ) ; ylabel ( Number of Sick INdividuals ) ; hold o f f ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Routine parameters.m generates host v i r a l dynamical parameters % for each individual in the population using 1 patients data % from the paper Stafford et. al (2) function [ d, lambda, k, delta, Pi, c ] = parameters (n ) ; Dynamics at the Horsetooth 11 Vol. 2, 21

12 % Stafford parameter d (TW mu) meand=.189; stdevd =.5727; for i =1:n d( i )=tuck ( [ meand stdevd ] ) ; % Stafford parameter lambda (TW s ) meanlambda=.189; stdevlambda =.5727; for i =1:n lambda( i )= tuck ( [ meanlambda stdevlambda ] ) ; % Stafford parameter k (TW k) meank=.1179; stdevk =.1482; for i =1:n k( i )= tuck ( [ meank stdevk ] ) ; %Stafford parameter delta (TW a) meandelta =.366; stdevdelta =.193; for i =1:n delta ( i )= tuck ( [ meandelta stdevdelta ] ) ; %Stafford parameter pi (TW c ) meanpi=1427; stdevpi =249; for i =1:n Pi ( i )= tuck ( [ meanpi stdevpi 98 71]); % Stafford parameter c (TW gamma) meanc=3; stdevc =; for i =1:n c ( i )= tuck ( [ meanc stdevc 4 ] ) ; % System has two fixed points P1 and P2 % Fixed point P1 : prob = ; for j =1:n z ( j)= ( Pi ( j) delta ( j )) k( j ) lambda( j )/(d( j ) delta ( j ) c ( j ) ) ; i f z ( j ) > 1 prob = prob+1; Dynamics at the Horsetooth 12 Vol. 2, 21

13 figure ( ) ; probp1sadd = prob/n probp1node = 1 probp1sadd x = [ probp1sadd probp1node ] ; xd={ SADDLE, STABLE NODE } ; nx=numel(x ) ; bar (x, g ) ; set ( gca, ylim, [, 1. 2 ] ) % xticklabel, xd ) ; text (1: nx, repmat (1.1,1, nx ),xd,... horizontalalignment, center,... fontsize, 1 2,... fontweight, bold ) ; xlabel ( Type of C r i t i c a l Point ) ; ylabel ( Frequency ) % Plotting positions of P1 for each individual in the population for i =1:n x1( i ) = lambda( i )/d( i ) ; figure ( ) ; plot3 (x1,,, b ); grid on axis square xlabel ( T ) ; ylabel ( T ); zlabel ( V ) ; % Fixed point P2 : for j =1:n x2( j ) = c ( j ) delta ( j )/( k( j ) ( Pi ( j) delta ( j ) ) ) ; y2( j ) = lambda( j )/ delta ( j) d( j ) c ( j )/( k( j ) ( Pi ( j) delta ( j ) ) ) ; z2 ( j ) = lambda( j ) ( Pi ( j) delta ( j ))/( c ( j ) delta ( j )) d( j )/k( j ) ; sigma ( j ) = delta ( j)+c ( j)+d( j)+k( j ) ( x2( j)+z ( 2 ) ) ; dl ( j)= ( delta ( j)+c ( j )) ( d( j)+k( j ) z2 ( j ))+d( j ) k( j ) x2( j ) ; eps ( j)= delta ( j ) c ( j ) k( j ) z2 ( j ) ; y=[1 sigma ( j ) dl ( j ) eps ( j ) ] ; p=roots (y ) ; pp( j)=p ( 1 ) ; qq( j)=p ( 2 ) ; rr ( j)=p ( 3 ) ; ssadd = ; snode = ; sfocus = ; for m = 1:n i f ( real (pp(m))<) && ( real (qq(m))<) && ( real ( rr (m))<) i f (imag(pp(m)) == ) && (imag(qq(m)) == ) snode = snode+1; e l s e sfocus = sfocus +1; e l s e ssadd = ssadd+1; Dynamics at the Horsetooth 13 Vol. 2, 21

14 % Verifying that sigma dl eps > for any individual for j j =1:n ; hh( j j ) = sigma ( j j ) dl ( j j ) eps ( j j ) ; indneg = find (hh <= ) ip =; for mm=1:n ; i f ( real (qq(mm)) real ( rr (mm))) == ip=ip +1; ip probp2sadd = ssadd/n probp2node = snode/n probp2focus = sfocus /n probp2 = probp2sadd+probp2node+probp2focus figure ( ) ; xx = [ probp2sadd probp2node probp2focus ] ; xxd={ SADDLE, STABLE NODE, STABLE FOCUS } ; nxx=numel(xx ) ; bar (xx, g ) ; set ( gca, ylim, [, 1. 2 ] ) ; text (1: nxx, repmat (1.1,1, nxx ), xxd,... horizontalalignment, center,... fontsize, 1 2,... fontweight, bold ) ; xlabel ( Type of C r i t i c a l Point ) ; ylabel ( Frequency ) % Plotting positions of P2 for each individual in the population figure ( ) ; plot3 (x2, y2, z2, b ); grid on axis square xlabel ( T ) ; ylabel ( T ); zlabel ( V ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Parameter calculation based on normal distribution function [ u]= tuck (x ) ; mu = x ( 1 ) ; sigma = x ( 2 ) ; Dynamics at the Horsetooth 14 Vol. 2, 21

15 min = x ( 3 ) ; max = x ( 4 ) ; R=rand ; u = sigma sqrt (2) erfcinv (R ( erfc ((mu max)/( sigma sqrt (2)))... erfc ((mu min)/( sigma sqrt (2))))+ erfc ((mu min)/( sigma sqrt (2))))+mu; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dynamics at the Horsetooth 15 Vol. 2, 21

16 References [1] Stafford, M.A., Corey, L. & Cao, Y. et al., 2 Modeling plasma virus concentration during primary HIV infection J. Theor. Biol. 23, [2] Tuckwell, H.C., Toubiana, L., 27 Dynamical modeling of viral spread in spatially distributed populations: stochastic origins of oscillations and density depence Biosystems 9 (2), [3] Tuckwell, H.C., Wan, F.Y.M., 2 Nature of equilibria and effects of drug treatments in some simple viral population dynamical models IMA J. Math. Appl. Med. Biol. 17, Dynamics at the Horsetooth 16 Vol. 2, 21