Modeling epidemics with differential equations

1 Modeling epidemics with differential equationsRoss Beckley1, Cametria Weatherspoon1, Michael Alexander1,Marissa Chandler1, Anthony Johnson2, and Ghan S Bhatt11 Tennessee State University,2 Philander Smith 21, well known SIR models have been around formany years. Under some suitable assumptions, the models pro-vide information about when does the epidemic occur and when itdoesn t. The models can incorporate the birth, death, and immu-nization and analyze the outcome mathematically. In this projectwe studied several SIR models including birth, death and immu-nization.

2 We also studied the bifurcation analysis associated withthe disease free and epidemic on some mathematical assumptions, it is known that epi-demics can be modeled mathematically in order to study the severityand prevention mechanism. This model (SIR) is used in epidemiologyto compute the number of susceptible, infected, and recovered peoplein a population at any time. It can be used to explain the change in thenumber of people needing medical attention during an epidemic. Thewhole population is divided into three classes,S;the number of suscep-tible,I;the number of infected andR;the number of recovered duringan epidemic.

3 This model assumes that the total population remainsthe same with closed demography meaning that there is no birth andno natural death. Any disease related death, however, can be includedinR:We study the basic SIR model with some reasonable we include herd immunity, birth and death into the model. Theconstant vaccination at birth is also considered . The ultimate goal isto model the issue of saturated susceptible population, the time delayof infected to become infectious, the stability of equilibrium solutionsand associated nition individuals are individuals that have neverbeen infected and they are able to catch the disease.

4 Once they have it,moving into the infected compartment. Infected individuals can spreadthe disease to susceptible individuals. Recovered individuals in the re-covered compartment are assumed to be immune for (t) be the number of susceptible individualsI(t) be the numberof infected individuals and letR(t) be the number of recovered indi-viduals at timetrespectively. It is also assumed thatS+I+R=N:Also we normalize this sum by dividing each of the variables byN:Westill denote the new variables by the same lettersS; SIR ModelsSIR models have been around for many years, for example [3, 5, 4,2, 6] and the references there in.

5 The first one was introduced andpublished in 1927, in Contribution to the Mathematical Theory ofEpidemics , written by William Kermack and Anderson introduced the important compartments, which make up the SIRmodel, S- susceptible, I - infected and R - recovered. They searchedfor a mathematical answer as to when the epidemic would terminateand observed that, in general whenever the population of susceptibleindividuals falls below a threshold value, which depends on severalparameters, the epidemic Basis population is fixed soS+I+R= 1:The disease spreads through the interaction of susceptible and assume that only a fraction of this interaction causes the diseaseto pass from an individual (I) to a susceptible individual (S.)

6 So therate of change ofSis proportional to the product ofSandI:Weassume that the individuals recover at a rate of so the period ofinfection is1 days. The only way a person can leave the susceptiblegroup is to become infected. The only way a person can leave theinfected group is to become recovered. Once a person is recovered, theperson is no longer susceptible and is immune. Age, sex, race and socialstatus do not affect the probability of a person being affected. Thereis no inherited immunity at this time. The people of the populationmix homogeneously.

7 Based on the above assumptions the differential3equations governing the disease can be modeled asdSdt= SIdIdt= SI I(1)dRdt= the total population is assumed to be constant, thethird equation can be derived from the first two. Basically we studythe first two in turns out that the epidemic occurs ifdIdt>0;it doesn t ifdIdt<0:Sofor the epidemic to occur we have to have S > implyingS > :Forthe epidemic to terminate the rate of change ofIhas to be negative,this implies thatS < :The phase portrait Figure 1 shows this Phase Portrait of SIR modelDe nition 2(Basic Reproductive Number).

8 The basic reproductivenumberR0(the average number of persons infected by one case in atotally susceptible population in absence of interventions aimed at con-trolling the infection). SinceS= 1initially, the ratio S = =R0:This is one of the most important parameters in the SIR Modeling ofany especially important in this case as it will informone as to when an epidemic is in progress. So ifR0>1 an epidemicwill occur and ifR0<1 there will be no epidemic. The values ofR0areknown for various diseases. For example for Swine flu, it is reported tobe 1:3 1:6 in [1]The first two equations can be solved forIandSas in [3] Thevariation ofIversusScan be seen from the figure provided Figure solutions be written as, [3].

9 (2)I(S) = S+1R0lnS+ 1:The graphs of this equation are shown for different values ofR0:Thesystem of equations can be solved for several values of the graphs ;for different values ofR0 The typical solutions of the above equations are shown in Figure3,using general solutions over this portion of the model we usepto bethe proportion of susceptible population that is immunized before theoutbreak of an epidemic and assume the above mentioned conditions,new equations governing the disease can be written = (1 p)SI(3)I = (1 p)SI IAn outbreak of the epidemic mathematically means thatI >0) (1 p)SI I >0) (1 p)S > )( = )(1 p)S >1)R0>11 value ofR0is apprx.

10 1:6 for Swine flu [1]. The aboveinequalities says that at least 38% need to be immunized in order tocontain the with birth and a modification to the SIR modelwe introduce birth and death. We assume that all death is natural. Thevariablemis used to represent a constant rate of birth and death. Thebasic reproduction number is now given byR0= + new equa-tions with the consideration of birth and death are:Figure possible solution curves for a particular disease5dSdt=m I mS(4)dIdt= IS (m+ )IThe system of equations have two equilibrium free equilibrium,(S1; I1) = (1;0)and theepidemic equilibrium,(S2; I2) =( +m ;m (R0 1)):The eigenvalues of the Jacobian matrix reveal the stability of theseequilibrium solutions.