In this section, we propose a numerical scheme for our model and gives some illustra-tions.
We adopt a finite differences scheme which is progressive of order 1 in time and regressive of order 1 in age. Our model has a structure of the following partial differential equation on the real axe:
∂u
∂t +∂u
∂a=f(t, a). (54)
For equation (54), the numerical scheme is defined by:
un+1i −uni
∆t +uni −uni−1
∆a =f(tn, ai); (55)
where i and n are the index of age and time discretization respectively; and uni :=
u(tn, xi).
We recall that, generally, all explicit numerical scheme is conditionally stable (Stricwerda[45]).
To ensure the stability of the scheme (55) the necessary condition is the famous Courant-Friedrichs-Lewy (CFL) condition given as follow:
∆t
∆a 61. (56)
For a given age step discretization∆a, the restriction∆t6∆ais necessary for the time step discretisation∆t.
We are able now to give the solution of the problem (1)-(2) on some time interval [0, T]using the above numerical scheme.
The age-specific reproduction ratef(a)is taken to be f(a) =
(1
5sin2π(a−15)
30
if 15≤a≤45;
0if not.
The fecundity functionf(.)is stated here in units of 1 / years for easier readability and assumes that from age15to45years a woman will generally give birth to three children, sinceRa+
0 f(a)da= 3,wherea+= 80is the largest age allowed for the simulation.
We also consider a low value of recruitmentΛ(.) Λ(a) =
(1
10sin2
π(a−17) 43
if 17≤a≤60;
0if not.
This recruitment assume that the total number of recruitment at timetis approximately equal two, that isRa+
0 Λ(a) = 2.15
The transmission coefficientβ(., .)is assume to be
β(a, s) =
β0sin2
π(a−14) 46
sin2
π(s−14) 46
, ifa, s∈[14,60];
0if not.
0 20
40 60
80
0 20 40 60 80
0 0.5 1
x 10−3
Age (year) Age(year)
Transmission rate
(a)
0 20 40 60 80
0 0.05 0.1 0.15 0.2
Age (year)
Reproduction rate at age a
(b)
Figure 1: (1a) Transmission coefficientβ(., .)when the transmission constantβ0= 10−3. (1b) Fecundity functionf(.).
Table 1: Numerical values for the parameters of the model Parameters Description Estimated value β0 Transmission constant Variable p Vertical tranmission rate Variable
µ Natural death rate 0.0101/yr1
r Rate of effective therapy 1/yr1 φ Rate at witch infectious 0.75/yr1
become loss of sight
γ Rate at witch lost of sight 0.02/yr1 return to the hospital
d1 Death rate of infectious 0.02/yr1 d2 Death rate of lost of sight 0.2/yr1
Note: Source of estimates.
1Assumed.
wherein the nonnegative constantβ0 (transmission constant) will be variable. Figure1 illustrates the transmission coefficientβ (forβ0 = 10−3) and the fecundity functionf. The other parameters of our system are arbitrarily chosen (see Table1).
We provide numerical illustrations for different values of vertical transmissionp:0.02, 0.2and0.5
In Figure2, the vertical transmission rate of the disease is fixed to bep= 0.02. We observe that infectious individuals (infected and lost of sight) are between 17 and 70 of age. The number of young infectious (namely infectious with agea < 17) is negligible, because the value of vertical transmission ratepis low.
In figure3, the vertical transmission rate of the disease is fixed to bep = 0.2. We observe that much of the infectious individuals (infected and lost of sight) are between
17 and 70 of age. Let us also observe that the number of infectious individuals with age between 17 and 70 is approximately the same than the number of infectious individuals with age between 17 and 70 whenp = 0.02(see Figs 2-3). But now, there are also infectious individuals with agea <17which was not the case whenp= 0.02.
The same observation is given by Figure4where the vertical transmission rate of the disease is fixed to bep= 0.5. Hence Figures2-4emphasize that the vertical transmission of the disease really has an impact on the dynamics and the spread of the disease into the host population. See also Table2for the impact of the vertical transmission of the disease on the spread of the epidemic.
0 20 40 60 80 0
20 40
60 80 0
50 100
Time(year) Infected
Age(year)
Cases
(a)
0 20
40 60
80 0 20 40 60 80
0 50 100
Age(year) Lost of sight
Time(year)
Cases
(b)
0 10 20 30 40 50 60 70 80
1 2 3 4 5 6 7 8
Infected newborn
Time(year)
Cases
(c)
0 10 20 30 40 50 60 70 80
0 5 10 15 20 25 30
Population after 15 years
age(year)
Cases
infected Lost of sight
Total cases (I): 96.043 Total cases (L): 875.52 Total cases (I+L): 971.56
(d)
Figure 2: The transmission constant and the vertical transmission rate are fixed to be β0 = 10−3andp= 0.02. The other parameters are given by Table1. (2a) Distribution of Infected individuals. (2b) Distribution of Lost of sight. (2c) Distribution of infected newborn. (2d) Distribution of Infected and Lost of sight individuals after80years of time observation.
0 20 40 60 80 0
20 40
60 80 0
50 100 150
Time(year) Infected
Age(year)
Cases
(a)
0 20
40 60
80 0 20 40 60 80
0 50 100
Lost of sight
Age(year) Time(year)
Cases
(b)
0 10 20 30 40 50 60 70 80
0 20 40 60 80 100 120
Infected newborn
Time(year)
Cases
(c)
0 10 20 30 40 50 60 70 80
0 5 10 15 20 25 30
Population after 15 years
age(year)
Cases
infected Lost of sight
Total cases (I): 125.92 Total cases (L): 996.04 Total cases (I+L): 1122
(d)
Figure 3: The transmission constant and the vertical transmission rate are fixed to be β0 = 10−3andp= 0.2. The other parameters are given by Table1. (3a) Distribution of Infected individuals. (3b) Distribution of Lost of sight. (3c) Distribution of infected newborn. (3d) Distribution of Infected and Lost of sight individuals after80years of time observation.
Table 2: Impact of the vertical transmission of the disease.
Vertical transmission rate (p) Rate increase over the case whenp= 0
p= 0.02 1.8%
p= 0.2 17.5%
p= 0.5 43.8%
Total cases (I+L) whenp= 0: 954.85 cases. (i.e. when the vertical transmission of the disease is neglected in the host population.
0
Population after 15 years
age(year)
Cases
infected Lost of sight
Total caases (I): 175.72 Total cases (L): 1196.9 Total cases (I+L): 1372.7
(d)
Figure 4: The transmission constant and the vertical transmission rate are fixed to be β0 = 10−3andp= 0.5. The other parameters are given by Table1. (4a) Distribution of Infected individuals. (4b) Distribution of Lost of sight. (4c) Distribution of infected newborn. (4d) Distribution of Infected and Lost of sight individuals after80years of time observation.
7. Conclusion
In this paper, we consider a mathematical model for the spread of a directly transmit-ted infections disease in an age-structured population with demographics process. The disease can be transmitted not only horizontally but also vertically from adult individuals to their children. The dynamical system is formulated with boundary conditions.
We have described the semigroup approach to the time evolution problem of the ab-stract epidemic system. Next we have calculated the basic reproduction ratio and proved that the disease-free steady state is locally asymptotically stable ifR0 <1, and at least one endemic steady state exists if the basic reproduction ratioR0is greater than the unity.
Moreover, we have shown that the endemic steady state is forwardly bifurcating from the disease-free steady state atR0 = 1. Finally we have shown sufficient conditions which guarantee the local stability of the endemic steady state. Roughly speaking, the endemic
steady state is locally asymptotically stable if it corresponds to a very small force of in-fection.
However the global stability of the model still an interesting open problem. Moreover, biologically appropriate assumptions for the unique existence of an endemic steady state is also not yet know.
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