Transcript Lab4
We will create Mathematical codes in Vcell for 1. SIR Model 2. Fitzhugh-Nagumo Model SIR Model : The population can be subdivided into a set of distinct classes, dependent upon their experience with respect to the disease. These are Susceptible, Infectious or Recovered. Thus comes the term the SIR model. Rate of transmission-- Modeling Assumptions: 1. Non-lethal epidemic (i.e. S+I+R= N=constant) 2. Once a person recovered, can not get infected dS = rate of change of suceptible = - (infection rate)*S, dt Infection rate depends on number of infectious individuals (I) and the Probabily to catch the infection (b). The equation for susceptible is- dS b * I * S dt Rate of change of infectious population- dI dS g*I dt dt , g is the recovery rate Equation for infectious population - dI b*I *S g *I dt Rate of change of recovered population- dR g*I dt Initial condition: S(0)=N-I0 I(0)=I0 R=0. S+I+R= N= constant Start Math Model in Vcell FileNewMathModelNon-spatial Start writing code here First declare the variable parameters for example S_init, I_init, R_init ( s=0.9, I=0.1 R=0) Then constants, b and g (2.0 and 1.0 ) Declare VolumeVariable S, I and R Declare 3 Function Susceptible, Infectious, Recovered populations Write 3 ODEs as before Finally the Model will be like this. Run simulation and see the results S(0)=0.995; I(0)=0.005, b=2.0, g=1.0 Time=20 sec dI/dt = b*S*I − g*I= I (b*S − g) S(0)>(g/b), dI/dt >0 – epidemics S(0)<(g/b), dI/dt<0 – no epidemics S(0)=9.0,I(0)=0.1,b=g=1.0 S(0)=0.9, I(0)=0.1, b=g=1.0 Fitzhugh-Nagumo Model for Neural Impulses: This Model is a simplification of the Hodgkin-Huxley Model. 1. Propagation of nerve signal is electrical in nature. 2. The signal propagates down the length of the axon to synapses, which are connected to the neighboring neurons. 3. Propagated signal is called action potential 4. Neuronal signals travel along the cell membrane of the axon in the form of local voltage difference accross the membrane. Reference: http://www.scholarpedia.org/article/FitzHugh-Nagumo_model V – Excitability of the system (Membrane potential in the axon) C – ‘openness’ of ion channels, representing combined forces that tend to return the state of the axonal membrane to rest I – externally applied voltage or stimulus that leads to an excitation dV I V 0.2 V V 1 C dt dC 0.002* V C dt Start writing MathModel in Vcell Constant declaration V_init C_init I Try, V_init=0.1 C_init=0; I=0; VolumeVariable V VolumeVariable C Function J1 Function J2 I + V* (0.2-V)* (V-1) + C ; 0 .002*(V-C) ; ODEs as before Simple MathModel Run simulation: Try initially, I=0; C=0 and V= 0.1; t_end= 200 sec I=0; C=0 and V= 0.3; t_end= 800 sec Exercise 1: play with this Fitzhugh-Nagumo model in VC; keep all parameters parameters the same as in the previous slide: start increasing I to 0.05, then 0.1, then 0.15, then 0.2 Think about what is going on! Try to plot V(C) and think some more… dV I V 0.2 V V 1 C dt dC 0.002* V C dt Run the simulation for t_end= 2000 sec C-V curve for different values of I., Spatial diagram C and V w.r.t. time for different values of I. Exercise 2 Consider SIR model in more detail. dS B dt birth or immigration SI S getting sick natural death with rate alpha dI SI I I I dt getting sick natural death recovering death from decease with rate alpha dR I R dt recovering natural death with rate alpha B=20; alpha=1; beta=0.1; gamma=1; lambda=1; with rate lambda Exercise1 This is my math model. In my model— Beta=b Alpha= a Gamma=g Lambda=l S(0)=10, I(0)=1, b= 0.1 For b=0.2 Seasonal getting sick with period 1 year (for example Flu ) Take parameters-B=20; alpha=1; beta=0.1; gamma=1; lambda=1; dS B dt birth or 0 1 cos t SI S getting sick natural death with rate alpha immigration dI 0 1 cos t SI I I I dt natural death death from decease recovering getting sick with rate alpha with rate lambda dR I R dt recovering natural death with rate alpha Try to play with this model in VC; vary parameters; think about what is going on Something new-- Here beta is not constant, rather it fluctuates periodically. beta = beta_0 * (1+cos t) So we have to declare this periodic function. HOW ??? Write, Function K cos (t); Use this function to write other functions Rest of the code is same as before !!!!! My MathModel looks like this... For small b=0.1, almost no infection For b=0.2 S(0)=10; I(0)=1; b=0.3, l=g=a=1.0