Simple Infection Model
Huaizhi Chen
Simple Model

The simple model of infection process
separates the population into three classes
determined by the following functions of age
and time:



X(a,t) = susceptible population
Y(a,t) = infected population
Z(a,t) = immune population
Governing Equations

The dynamics of the simple model is governed by the
following partial differential equations:
X / t  X / a  ( (t )   (a)) X (a, t )
Y / t  Y / a   (a) X  ( (a)   (a)  (a))Y (a, t )
Z / t  Z / a  (a)Y   (a)Z (a, t )
Parameters

Where
 (a)  age-specific host death rate, per capita
 (a)  per capita recovery rate
 (a)  per capita disease-induced death rate
 (a)  "force of infection" at time t
Boundary Conditions


We have the following initial conditions:
At a = 0
–

Y(t)=Z(t)=0, X(t) = B(t), where B(t) is net birth rate
at t
At t = 0
–
X(a), Y(a), Z(a) must be defined.
Additional Classes

Other Classes Can be Incorporate into the
model
–
Take Latent Period:


We can divide the Y period to H and Y’
Where H is the latent period and Y’ is the infectious
period
Latent Period
X / t  X / a  ( (t )   (a)) X (a, t )
H / t  H / a   X  (   (a)) H (a, t )
Y '/ t  Y '/ a   H  (    )Y '(a, t )
Z / t  Z / a  (a)Y   (a)Z (a, t )

Where we have a new parameter
representing the per capita transfer from
latent infected to infectious infected.
Maternal Antibodies

Temporary immunity can be granted to newly
born infants from an immune mother. This
can be incorporated into the simple model.
Verticle Transmission

Some infections can be passed directly to the
new-born offspring of an infected parent.

This phenomenon can be represented by
tweaking the boundary condition at a = 0 and
have X(0,t) = B1(t), Y(0,t) = B2(t), and Z(0,t) =
0.
Separate Treatments of Male/Female

Often, for sexually transmitted diseases, it
would be helpful to stratify the variables by
sex. Like - Xm, Xf, Ym, Yf, Zm, Zf and govern
those classes with separate dynamics.
Recovery



v(t) can be modeled in various manners.
Type A (constant) and Type B (step)
Proportion Infected
Type
Type
B recovery
1
A recovery
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1
2
3
4
5
1
2
3
4
5
Loss of Immunity

We can also modify the simple model to incorporate
loss of immunity by incorporate the parameter
gamma.
X / t  X / a   Z  ( (t )   (a)) X (a, t )
Y / t  Y / a   X  ( (a)   (a)  (a))Y (a, t )
Z / t  Z / a  (a)Y  ( (a)   (a))Z (a, t )
Natural Mortality


Mortality can be modeled similarly to
recovery.
Type I and Type II
Type
Type
I Mortality
1
1
0.8
0.8
0.6
0.6
0.4
0.4
II Mortality
0.2
0.2
1
2
3
4
5
1
2
3
4
5
Disease-Induced Mortality


Generally a constant is used in place of the
alpha function; however depending on the
disease, it may be advantageous to model
alpha by age.
For example, malaria and measles exhibit
greater mortality in infants.
Transmission

The transmission parameter can be modeled by:

    (a, a ')Y (a, t )da
0


Where beta is the probability that an infected
individual of age a would infect a susceptible of age
a’.
A specific case with constant probability is

    Y (a, t )da
0
Other Concerns

Seasonality
–

Nutritional State
–

The beta parameter in the transmissions equation
can be very seasonal.
Nutritional state of the population can exert
changes in the mortality, transmission, and other
rate parameters.
Homogenous Mixing
–
Examples have shown important effects of
heterogeneity in most real populations.