Mathematical model.
In 1985, Kallen, Arcuri and Murray (377-393) presented a simple mathematical model for the spread of the rabies epidemics to approximate the minimum width of a break and enable the control of the spatial progression of rabies.
Features and assumptions of the model.
The model developed by Kallen, Arcuri and Murray (377-393) was based on a deterministic approach where the fox populations were categorized into two groups, namely infective and susceptible foxes. In this classification, the infective foxes included those with rabies virus present in their saliva while the susceptible ones included those without traces of rabies virus in their saliva. One of the assumptions of the model was that transmission of the rabies virus happened solely through fox bite by a rabid fox with rabies virus in their saliva (Kallen, Arcuri and Murray 379). As a result, for transmission of the rabies virus between foxes, a physical contact of a rabid fox with a susceptible fox is necessary where the rabid fox bites the susceptible one.
The second assumption made during the development of the model was that the virus did not cause variable fatalness in foxes and therefore such a factor as immunity did not offer a higher survival advantage of some foxes over others. Another assumption of the model is that fox populations occur in territorial communities with non-overlapping home ranges where members of a territory maintain strict boundaries. However, in the case of a rabies infection, a fox develops behavioral dysfunction that causes them to become aggressive as well as lose the sense of territoriality and direction leading to random movement within and outside their territory. However, for such behavioral alteration to happen the rabies virus must enter the central nervous system to induce the behavioral change (Kallen, Arcuri and Murray 379).
Development of the mathematical model.
Based on the four assumptions observed, it is possible to develop a simple equation to model the change of number of the infective foxes per a unit time, say one year (Kallen, Arcuri and Murray 380-382). In this case, the numerical change of the population of the infective foxes in a territory can be hinted by the following expression.
In a short period interval, the infective populations’ rate of loss is given by the susceptible population’s temporal variation during the same interval. As a result, random walks can be used for valid approximation of the infective population dispersal through the following model equations.
ƏS/ Ət=-KIS
ƏI/ Ət=DƏ2I/ Əx2 + KIS- µI (i)
Where,
S = susceptible population density.
I =Infective population density.
X=one dimensional spatial problem.
K=transmission coefficient.
1/µ=life expectancy of the infective fox.
T=time.
D=Diffusion coefficient.
The diffusion coefficient is derived by
D=kA (ii)
Where,
K=rate of dissemination of infective foxes.
A=average area of the territory.
For the derivation of a functional equation for a rabies epidemic, three additional assumptions were made. One of the assumptions was that the susceptible populations remained static due to the balancing effects of births. The second assumption was that migration was absent among all susceptible foxes regardless of their sex and stage of life. The third assumption was that the land covered by a fox territory was even and homogenous to avoid preferred directions of migration (Kallen, Arcuri and Murray 380-382).
U = I/S0
v = S/S0,
X*= KS0D12( x),
T*=KS0t
R=µ/KS0
Where,
S0 = initial susceptible density.
After dropping the asterisks for convenience, the model equations can be rewritten as:
Əu/tƏ = Ə2u/ Əx2 + u(v-r)
The simplified equation, Əv/ Ət = -uv. (iv)
Where
u≥0, and 0<v≤1.
Based on this information, the critical minimum fox density (Sc) can be given by:
Sc = µ/K =rS0 (v)
Then after an infective wave, the number of susceptible foxes (a) in a territory is within the range of 0<a<r<1, and it can be expressed in an equation.
a-r log a=1 (vi).
If two species travel together and at the same speed, their waveforms are bound by the following equation.
C= 2√(1-r) (vii)a.
In dimensional terms, the waveforms of two species are bound by dimensional terms in the following equation.
C=2[D(KSo - µ)]1/2 = 2[Dµ(1/r-1)]1/2 (vii)b.
Epidemiological parameters.
Parameters r and a are arrived at through the existing literature where at the highest bound of the mortality rate of foxes is assumed because of the high mortality rate related to the epizootic fronts. If the upper bound o the mortality rate is 80 percent, the surviving proportion of foxes (a) is equal to 0.2. However, if the rate of population change o the susceptible foxes is 50 percent, then r=0.5 (Kallen, Arcuri and Murray 384-385). For estimation of µ, it must be recalled that 1/µ represents the life expectancy of an infective fox. An average life expectancy of an infective fox is about 35 days, and thus µ is approximately 10yr-1. Moreover, D was estimated by D=kA, where k was approximately 12yr-1 and the average area of a territory (A) was assumed to be 5 Km2. The primary use of this model was the prediction of the spread of an epizootic epidemic from a known dispersal point.
Works Cited.
Källén, A, P Arcuri, and JD Murray. "A Simple Model for the Spatial Spread and Control of Rabies." Journal of Theoretical Biology. 116.3 (1985): 377-93
