Epidemic models with an infected-infectious period

Geographic spread of epidemics was studied by a pion ing work of Noble@1#, but it is less understood and less we studied than its temporal evolution. However, recent wo such as@2# and @3#, for instance, propose new approach taking into account the role of cross-diffusion or a variab population size. Here we consider a simple SI ~susceptibleinfectious! model that leads to a very good velocity of spre of epidemics in accordance with the experimental results tained for the Black Death catastrophic plague pandem This model is compared with a model of three species. Our main assumption is to consider a characteristic tim of delay in the appearance of the infectious members, wh measures the period between the infected-infectious tra tion. When a susceptible population is infected, there i time t.0 during which the infectious agents develop with the susceptible individual organisms and it is only after t time that the infected population becomes itself infectio ~or infective!. The corresponding model mechanisms for t development and spatial spread of the disease are phe enologically derived. The traveling wave analysis of t model is carried out and the asymptotic velocity for an fectious solitary wave is found and it is compared with t older results of Noble.


I. INTRODUCTION
Geographic spread of epidemics was studied by a pioneering work of Noble ͓1͔, but it is less understood and less well studied than its temporal evolution.However, recent works such as ͓2͔ and ͓3͔, for instance, propose new approaches taking into account the role of cross-diffusion or a variable population size.Here we consider a simple SI ͑susceptible-infectious͒ model that leads to a very good velocity of spread of epidemics in accordance with the experimental results obtained for the Black Death catastrophic plague pandemic.This model is compared with a model of three species.
Our main assumption is to consider a characteristic time of delay in the appearance of the infectious members, which measures the period between the infected-infectious transition.When a susceptible population is infected, there is a time Ͼ0 during which the infectious agents develop within the susceptible individual organisms and it is only after that time that the infected population becomes itself infectious ͑or infective͒.The corresponding model mechanisms for the development and spatial spread of the disease are phenomenologically derived.The traveling wave analysis of the model is carried out and the asymptotic velocity for an infectious solitary wave is found and it is compared with the older results of Noble.

II. THE FIRST MODEL
The SI model consists of only two populations, infectious I(x,t) and susceptible S(x,t), which interact.We model the spatial dispersal of the density of infectious individuals I and the density of susceptible individuals S by simple diffusion and consider the infectious and susceptible populations to be described by the same diffusion coefficient D .We consider the transition rate from susceptible to infected to be proportional to rSI, where r is a constant parameter.This means that rS is the number of susceptible individuals who catch the disease from each infectious unit.The susceptible members who catch the disease become infected members, an intermediate stage between susceptible and infectious.After a period , infected members become infectious and may transmit the disease.The parameter r measures the transmission efficiency of the disease from infectious to susceptible individuals.We assume that the infectious members have a disease-induced mortality rate ␣I, where 1/␣ is the life expectancy.The evolution equations for the susceptible and infectious populations take the form where Uϭ(S,I) T , and fϭ( f s , f I ) is given by Introducing the dimensionless variables I*ϭI/S 0 , S*ϭS/S 0 , t*ϭrS 0 t, and x*ϭͱ rS 0 D x,

͑1͒
where S 0 is a representative population, the evolution equation system is where aϭrS 0 and we have omitted the asterisks for notational simplicity.The dimensionless parameter is given by ϵ ␣ rS 0 .
We look for traveling wave solutions, in the usual way by setting zϭxϪct in Eq. ͑2͒ where c is the wave speed, which must be determined.This will represent a wave of constant shape traveling in the positive x direction.Substituting this into Eq.͑2͒ yields the ordinary differential system for I(z) and S(z), SЉϩcSЈϪISϭ0, ͑3͒ PHYSICAL

Application to the Black Death plague
In dimensional terms, the speed of the traveling waves, V say, is given by V min ϭ2ͱrS 0 Dc.͑7͒ In order to apply our model to the experimental results, we must know the value of .This value could be related to the incubation period of the disease but we have not yet established a direct correspondence.
In order to analyze our results we take the same approximate values for the parameters used by Noble.The susceptible population density is assumed to be S 0 Ϸ50/miles 2 , the diffusion coefficient is DϷ10 4 miles 2 /yr, the transmission coefficient is rϷ0.4 miles 2 /yr, and the life expectancy is about 3.5 weeks, so ␣Ϸ15/yr.With these parameters we obtain that the speed for the classical case (aϭ0) is 447.2 miles/yr, somewhat greater than the experimental results of 200-400 miles/yr quoted by Langer ͓4͔.If we take the infected-infectious period of two weeks (aϭ0.822),which seems to be reasonable, the asymptotic speed is V min , where c fulfills the equality in Eq. ͑6͒ as may be shown by using the steepest descent method of Kolmogorov.This yields, after numerical calculation, 281.7 miles/yr, which lies entirely within the experimental range.

III. THE SECOND MODEL
A second model which takes into account the infectedinfectious period may be developed by including a third species.Let S(x,t) be the number density of susceptible members, I ˆ(x,t) the number density of infected members, and I(x,t) the number density of infectious members.We assume in this section that the infected members have an infectious transition rate I ˆ/ where is the characteristic time of transition from infected to infectious or the infectedinfectious period and assume that all the susceptible members who catch the disease become infected members.By assuming Ficks's law for the diffusive spread of members, we get the following set of equations: Using now the dimensionless variables ͑1͒ we obtain ͑we omit asterisks for notational simplicity͒ If we fix the reference frame onto the moving front by using the transformation zϭxϪct, we obtain SЉϩcSЈϪISϭ0, ͑10͒ We define now the vector Uϭ(v,w,u… T so that Eq. ͑12͒ can be rewritten in the form where I is the unity matrix and By linearizing A we obtain the following characteristic polynomial In order to have real values in Eq. ͑13͒ it is necessary that , with Ͻ1.Note that the constraint Ͻ1 is recovered both from Noble's work and from the first model.

Application to the Black Death plague
In dimensional units the asymptotic velocity has the form Taking the same characteristic values of Noble and assuming ϭ2 weeks, we get Vϭ339.5 miles/yr which lies entirely in the experimental range 200-400 miles/yr.We have shown with these two models that the introduction of an infected-infectious period , which is reasonable from the practical point of view, of two weeks, leads us to a speed of the disease propagation which lies entirely in the experimental range.In both models the speed of the disease is lower than in the classical model (ϭ0) due to the infected-infectious period.Murray ͓5͔ excuses the bad theoretical result by arguing that the classical model ( aϭ0) is extremely simple and does not take into account the nonuniformity in population density, the stochastic elements, and so on.The fact is that, with a simple extension of the classical model, we are able to obtain two better results taking into account the infective-infectious period, which is also invoked in a recent work ͓6͔.
where we have expanded in Taylor series the terms S(x,tϪa) and I(x,tϪa) by assuming S and I infinitely derivable.The only homogeneous steady state is (0,S ˆ) where S may be any positive real value.The problem consists of finding the range of values of such that a solution exists with positive wave speed c and non-negative I and S such that ͓1͔ I͑Ϫϱ ͒ϭI͑ ϱ ͒ϭ0 and 0рS͑Ϫϱ ͒ϽS͑ ϱ ͒ϭ1.
where uϵI and vϵSϪS ˆ.The second equation for Eq.͑4͒ is uncoupled from v and may be analyzed separately.Its characteristic equation is 2 ϩcϪϩe ca ϭ0.͑5͒Since we require I(z)→0 with I(z)Ͼ0, I(z) cannot oscillate about Iϭ0, otherwise I(z)Ͻ0 for some z and therefore we must have real values for .In order to have two real solutions for Eq.͑5͒ it is necessary that the restriction