MATHEMATICAL ANALYSIS OF A MASS ACTION MODEL
BY
OCHOCHE J. M. and MADUBUEZE C.E.
Department of Mathematics/Statistics/Computer science.
Federal university of Agriculture, Makurdi, Nigeria.
ochoche.jeffrey@uam.edu.ng
Abstract: We presented and analyzed a mass action model with vital dynamics. This type of model is suitable
for many childhood diseases like Mumps, Rubella and other highly contagious diseases like influenza. We
showed that the region in which the model makes biological sense is positively invariant, this means that any
solution of the model with initial condition in this region remain in the region for all times. The model has two
equilibria, the disease – free equilibrium (DFE) and the endemic equilibrium. Using the concept of 𝑅0 , we
showed that the DFE is locally asymptotically stable provided 𝑅0 < 1, we similarly proved the global stability
of the DFE using a suitable Lyapunov function. Further we showed that the endemic equilibrium exist only
if𝑅0 > 1. Numerical simulation showed that the contact rate is an important parameter in the transmission
dynamics of mass action models.
Keywords: Endemic equilibrium, Incidence function, Invariant region, Mass action,
1.0 INTRODUCTION
The study of infectious diseases can no longer thrive without mathematical modeling. The robust capacity of
modeling as a tool for testing theories and simulations that produce near perfect results has made it an
indispensable tool in the study of infectious diseases. Infectious disease models has contributed to the design and
analysis of epidemiological surveys, suggest crucial data that should be collected, identify trends, make general
forecasts, and estimate the uncertainty in forecasts. On the other hand, infectious diseases remain the greatest
threat to human existence. The bubonic plague killed over 20% of the population of Europe over a seven year
period in the 1300s. The Great Plague of London, 1664–66 killed more than 75,000 of total population of 460,000.
The influenza epidemic of 1918 – 19 killed 25 million people in Europe. Medical advances in the 20 th and 21st
century have failed to provide a general solution to the spread of infectious diseases as diseases like HIV continue
to ravage human population throughout the world. From 1981, when the disease was first reported, to date, HIV
has killed more than 25 million people and more than 30 million people are presently infected.
Disease incidence refers to the infection rate of susceptible through their contact with infectives [1], it is the
number of new cases per unit time or the rate at which new waves of the disease appear. The choice of incidence
function in a mathematical model is very important since the dynamics of the epidemic is determined by how new
cases of infection are generated. The three incidence functions frequently used in deterministic mathematical
models are; the saturated incidence(
𝛽𝑆𝐼
1+𝛼𝐼
), the standard incidence (
𝛽𝑆𝐼
𝑁
) and the mass action incidence 𝛽𝑆𝐼.
The mass action incidence is given by 𝛽𝑆𝐼, where S,I denote the number of susceptibles and infectives respectively
and N is the total number of individuals in the population such that 𝛽𝑁 is the number of adequate contact
required for the transmission of the disease. The mass action incidence is appropriate when N is not too large [2]
since it assumes that the pattern of daily encounter is dependent on the size of the community which implies that
the contact rate is an increasing function of the population. The mass action incidence is density – dependent since
contact rate per infective is proportional to the density of the infectious host. Measles, Mumps, Rubella, Chicken
pox, Polio and Influenza are diseases that are commonly modeled using the mass action incidence. While the
choice of incidence function mostly depends on the disease being modeled, sometimes analytical tractability is
needed and hence the mass action incidence has also been used in modeling HIV [3-5]
1
In [6], the role of the choice of incidence function was investigated using a vaccine-induced backward bifurcation in
HIV models. Several examples are given where backward bifurcations occur using standard incidence, but not with
their equivalents that employ mass action incidence.
2.0 MODEL FORMULATION
We now present a mass action model with vital dynamics using the Susceptible-Infected-Recovered
approach (SIR). We shall show the positivity of solutions of the model and proof its local stability using the
linearization approach.
2.1 Variables of the Model
The variables of the model are defined below
𝑆(𝑡) = The number of susceptible individuals at time, t
𝐼(𝑡) =The number of infected individuals at time, t
𝑅(𝑡) = The number of recovered individuals at time, t
2.2
Parameters of the Model
The parameters of the model are defined below
𝑄0 =Birth rate
µ =
Natural death rate
𝛽 = The contact rate, defined to be the average number of effective contacts with other (Susceptible)
individuals per infective per unit time.
𝛼 = The rate at which an infectious individual recovered per unit time.
𝛿 =Measles induced death rate
2.3
Assumptions of the Model
The following assumptions are made in the model.
1. Individuals are born susceptible.
2. Infected individuals spread the disease to susceptible and remain in the infected class (in the period of
infectiousness) before moving into the recovered class.
3.
Individuals in the recovered class are assumed to be immune for life.
2.4
The Model Equation
The model equations for the study are given below
𝑑𝑆
= 𝑄0 − 𝛽𝑆𝐼 − 𝜇𝑆
𝑑𝑡
𝑑𝐼
= 𝛽𝑆𝐼 − (𝛿 + 𝛼 + 𝜇)𝐼 2.0
𝑑𝑡
𝑑𝑅
= 𝛼𝐼 − 𝜇𝑅
}
𝑑𝑡
2.5
Feasible Solution
The region in which the model makes biological sense is given by:
𝑄0
}
𝜇
From the model equations 3.1 it will be shown that the region is positively invariant.
Consider the steps below:
From the model equations, the total interacting population is given by
𝑁 = 𝑆 + 𝐼 + 𝑅
that is;
𝑑𝑁 𝑑𝑆 𝑑𝐼 𝑑𝑅
=
+ +
𝑑𝑡
𝑑𝑡 𝑑𝑡 𝑑𝑡
Ø = {(S, I, R) ∈ R3+ : 𝑆 + 𝐼 + 𝑅 = 𝑁 ≤
Therefore, adding the differential equations 3.1 we have
𝑑𝑁
= 𝑄0 − 𝜇𝑁
𝑑𝑡
Integrating
Therefore,
𝑑𝑁
𝑑𝑡
= 𝑄0 − 𝜇𝑁 has an integrating factor 𝑒 𝜇𝑡
2
𝑑𝑁 𝜇𝑡
𝑒 + 𝜇𝑁𝑒 𝜇𝑡 = 𝑄0 𝑒 𝜇𝑡
𝑑𝑡
Such that
(𝑁𝑒 𝜇𝑡 )′ ≤ 𝑄0 𝑒 𝜇𝑡
∫(𝑁𝑒 𝜇𝑡 )′ ≤ ∫ 𝑄0 𝑒 𝜇𝑡
𝑁𝑒 𝜇𝑡 ≤
𝑄0 𝜇𝑡
𝑒 +𝑐
𝜇
When 𝑡 = 0
𝑁(0) ≤
𝑄0
+ 𝑐
𝜇
𝐶 ≥ 𝑁(0) −
𝑄0
𝜇
Hence
𝑄0 𝜇𝑡
𝑄0
𝑒 + (𝑁(0) − )
𝜇
𝜇
𝑄0
𝑄0 −𝜇𝑡
𝑁 ≤
+ (𝑁(0) − ) 𝑒
𝜇
𝜇
𝑁𝑒 𝜇𝑡 ≤
𝑁 ≤ 𝑁(0)𝑒 −𝜇𝑡 +
So that as 𝑡 → ∞, 𝑁(𝑡) ≤
𝑄0
𝜇
𝑄0
(1 − 𝑒 −𝜇𝑡 )
𝜇
, This means that every solution with initial condition in Ø remains in Ø for all
𝑡 > 0 . Therefore in Ø,our model is biologically feasible, mathematically well posed and positively invariant
3.0 POSITIVITY OF SOLUTIONS
We shall now prove that all the variables in the model equation 2.0 are non-negative.
LEMMA 1
Let the initial data set be (𝑆, 𝐼, 𝑅) ≥ 𝑂 ∈ Ø, then the solution set (𝑆, 𝐼, 𝑅) (𝑡) of the equations 2.0 is
positive for all 𝑡 > 0
Proof:
from equation 1 in 3.1 if it is assumed that
𝑑𝑆
= 𝑄0 − 𝛽𝑆𝐼 − 𝜇𝑆 ≥ −(𝛽𝐼 + 𝜇)𝑆
𝑑𝑡
then
𝑑𝑆
𝑑𝑆
≥ −(𝛽𝐼 + 𝜇)𝑆 or
≥ −(𝛽𝐼 + 𝜇)𝑑𝑡
𝑑𝑡
𝑆
Integrating both side of the inequalities gives
𝑑𝑆
∫
≥ ∫ −(𝛽𝐼 + 𝜇)𝑑𝑡
𝑆
𝑙𝑛𝑆(𝑡) ≥ −(𝛽𝐼 + 𝜇)𝑡 + 𝑐
𝑆(𝑡) ≥ 𝑐𝑒 −(𝛽𝐼+ 𝜇)𝑡
when 𝑡 = 0, we have
𝑆(𝑡) ≥ 𝑆(0)𝑒 −(𝛽𝐼+ 𝜇)𝑡 ≥ 0
From second equation of 2.0
3
𝑑𝐼
= 𝛽𝑆𝐼 − (𝛿 + 𝛼 + 𝜇)𝐼 ≥ −(𝛿 + 𝛼 + 𝜇)𝐼
𝑑𝑡
Therefore
𝑑𝐼
𝑑𝐼
≥ −(𝛿 + 𝛼 + 𝜇)𝐼 or ≥ −(𝛿 + 𝛼 + 𝜇)𝑑𝑡
𝑑𝑡
𝐼
Integrating both sides of the equations gives
𝑑𝐼
∫ ≥ − ∫(𝛿 + 𝛼 + 𝜇)𝑑𝑡
𝐼
𝑙𝑛𝐼(𝑡) ≥ −(𝛿 + 𝛼 + 𝜇)𝑡
at 𝑡 = 0, we have
𝐼(𝑡) ≥ 𝐼(0)𝑒 − (𝛿+𝛼+𝜇)𝑡 ≥ 0 since (𝛿 + 𝛼 + 𝜇) > 0
From third equation of 2.0
𝑑𝑅
= 𝛼𝐼 − 𝜇𝑅
𝑑𝑡
𝜇𝑡
Wich has an integrating factor 𝑒 so
𝑑𝑅 𝜇𝑡
𝑒 + 𝜇𝑅𝑒 𝜇𝑡 = 𝛼𝐼𝑒 𝜇𝑡
𝑑𝑡
(𝑅𝑒 𝜇𝑡 )′ = 𝛼𝐼𝑒 𝜇𝑡
𝛼𝐼 𝜇𝑡
𝑅𝑒 𝜇𝑡 =
𝑒 +𝑐
𝜇
when 𝑡 = 0
𝛼𝐼
𝑅(0) =
+𝑐
𝜇
𝛼𝐼
or
𝑅(0) − = 𝑐
𝜇
𝛼𝐼
𝛼𝐼
𝑅𝑒 𝜇𝑡 =
𝑒 𝜇𝑡 + (𝑅(0) − )
𝜇
𝜇
𝛼𝐼
𝛼𝐼 −𝜇𝑡
𝑅(𝑡) =
+ (𝑅(0) − ) 𝑒
𝜇
𝜇
𝛼𝐼
−𝜇𝑡
−𝜇𝑡
𝑅(𝑡) = 𝑅(0)𝑒
+ (1 − 𝑒 ) > 0 since 𝜇 > 0
𝜇
hence, all variables are positive for 𝑡 > 0
4.0 EXISTENCE AND STABILITY OF DISEASE-FREE EQUILIBRIUM
The equilibrium points of the system (S.I.R) can be obtained by equating the rate of change to zero.
That is
𝑑𝑆
𝑑𝐼
𝑑𝑅
= = =0
𝑑𝑡
𝑑𝑡
𝑑𝑡
Equations 3.2 results to the following equations
𝑄0 − 𝛽𝑆𝐼 − 𝜇𝑆 = 0
𝛽𝑆𝐼 − (𝛿 + 𝛼 + 𝜇)𝐼 = 0
𝛼𝐼 − 𝜇𝑅 = 0
In the absence of the disease, only the first equation remains with the reduced form
𝑄0 − 𝜇𝑆 = 0
𝑄0
⇒ 𝑆0 =
𝜇
Therefore, 𝐸0 = (𝑆0 , 0,0) = (
𝑄0
𝜇
, 0,0)
4.1 Basic Reproduction Number
In the mathematical epidemiology an important concept is related to the basic reproduction number (𝑅0 ). It
is a measure of the potential for a disease to invade a population or die out when introduced. The basic
reproduction number, 𝑅0 is defined as the expected number of secondary infections produced by an index
case in a completely susceptible population [7]. 𝑅0 is a threshold parameter such that if the disease free
4
equilibrium is locally asymptotically stable, then the disease cannot invade the population and 𝑅0 < 1,
whereas if the number of infected individuals grows, the disease can invade the population and 𝑅0 > 1 [8,9]
The basic reproduction number (𝑅0 ) of the system 3.1 can be derived as follows:
𝐹 = (𝛽𝑆0 )𝐼
Where 𝑆0 =
𝑄0
𝜇
𝑉 = (𝜇 + 𝛿 + 𝛼)
𝑉 −1 =
1
(𝜇 + 𝛿 + 𝛼)
𝑅0 = 𝑠𝑝𝑒𝑐𝑡𝑟𝑖𝑎𝑙 𝑟𝑎𝑑𝑖𝑢𝑠 of FV −1
𝑅0 =
𝑅0 =
𝛽𝑆0
𝜇+𝛿+𝛼
𝛽𝑄0
𝜇(𝜇 + 𝛿 + 𝛼)
4.2 Local Stability Of The Disease Free Equilibrium
In this section, we study the local stability of the disease free eqiulibrium; it is determined by Jacobian matrix,
𝐽(𝑆, 𝐼, 𝑅) of system equation (3.1)
−𝛽𝐼 − 𝜇
𝐽(𝑆, 𝐼, 𝑅) = [ 𝛽𝐼
0
𝛽𝑆
𝛽𝑆 − (𝜇 + 𝛿 + 𝛼)
𝛼
−𝜇
𝑄0
𝐽 ( , 0,0) =
𝜇
0
[0
The Eigen values are 𝜆1 = 𝜆3 = −𝜇 and 𝜆2 = 𝛽
𝜆1 , 𝜆3 < 0 , 𝜆2 < 0 if 𝛽
𝑄0
𝜇
𝑄0
𝜇
𝛽
𝛽
𝑄0
𝜇
𝑄0
− (𝜇 + 𝛿 + 𝛼)
𝜇
𝛼
− (𝜇 + 𝛿 + 𝛼)
− (𝜇 + 𝛿 + 𝛼) < 0
or
𝛽
𝑄0
< (𝜇 + 𝛿 + 𝛼)
𝜇
or
𝛽𝑄0
<1
𝜇(𝜇 + 𝛿 + 𝛼)
or
𝑅0 < 1
Thus we have proved the following theorem.
5
0
0]
−𝜇
0
0
−𝜇]
THEOREM 1: (Local stability of 𝐸0 )
If 𝑅0 < 1, the disease free equilibrium of system (3.1) is locally asymptotically stable.
GLOBAL STABILITY OF THE DISEASE FREE EQUILIBRIUM
Consider the Lyapunov function 𝐿 = 𝐼
𝐿̇ = 𝐼 ̇ = 𝛽𝑆𝐼 − (𝛿 + 𝛼 + 𝜇)𝐼
= (𝛽𝑆 − (𝛿 + 𝛼 + 𝜇))𝐼
≤ (𝛽𝑆0 − (𝛿 + 𝛼 + 𝜇))𝐼
= (𝛽
𝑄0
− (𝛿 + 𝛼 + 𝜇)) 𝐼
𝜇
= (𝛿 + 𝛼 + 𝜇)(𝑅0 − 1)𝐼
≤ 0 for 𝑅0 < 1
Since all the model parameters are positive, it follows that 𝐿̇ < 0 for 𝑅0 < 1; 𝐿̇ = 0 only if 𝐼 = 0, hence 𝐿 is a
Lyapunov function on Ø. Therefore by the LaSalle’s invariance principle, every solutions to the model
equation 3.1, with initial conditions in Ø approaches 𝐸0 at 𝑡 → ∞. Thus we have proved the following theorem
THEOREM 2 (Global stability of 𝐸0 )
If 𝑅0 < 1, the disease free equilibrium of system (3.1) is globally asymptotically stable.
5.0 EXISTENNCE OF ENDEMIC EQUILIBRIUM
We now investigate the persistence of the disease in the population. At equilibrium all the model equation
equals zero, that is
𝑑𝑆 𝑑𝐼 𝑑𝑅
=
=
=0
𝑑𝑡 𝑑𝑡 𝑑𝑡
or
𝑄0 − 𝛽𝑆𝐼 − 𝜇𝑆 = 0
𝛽𝑆𝐼 − (𝛿 + 𝛼 + 𝜇)𝐼 = 0} 3.3
𝛼𝐼 − 𝜇𝑅 = 0
From the second equation of 3.3, we have
𝛽𝑆𝐼 = (𝛿 + 𝛼 + 𝜇)𝐼
or
𝑆∗ =
(𝛿 + 𝛼 + 𝜇)
𝛽
From the first equation of 3.3 we have
𝑄0 − 𝛽𝑆𝐼 − 𝜇𝑆 = 0
or
6
𝑄0 − (𝛿 + 𝛼 + 𝜇)𝐼 − 𝜇
𝐼∗ =
𝐼∗ =
(𝛿 + 𝛼 + 𝜇)
=0
𝛽
𝛽𝑄0 − 𝜇(𝛿 + 𝛼 + 𝜇)
𝛽(𝛿 + 𝛼 + 𝜇)
(𝑅0 − 1)𝜇(𝛿 + 𝛼 + 𝜇)
𝛽(𝛿 + 𝛼 + 𝜇)
𝐼 ∗ = (𝑅0 − 1)
𝜇
𝛽
From the third equation of 3.3
𝛼𝐼 = 𝜇𝑅
or
𝑅∗ =
𝛼𝐼 ∗
𝛼
= (𝑅0 − 1)
𝜇
𝛽
We have thus proved the following lemma
LEMMA 2
The endemic equilibrium state of the model given as
𝐸∗ = (
(𝛿+𝛼+𝜇)
𝛽
𝜇
𝛼
𝛽
𝛽
, (𝑅0 − 1) , (𝑅0 − 1) ) exist if and only if 𝑅0 > 1
5.1 GLOBAL STABILITY OF THE ENDEMIC EQIULIBRIUM
In this section, we study the local stability of the endemic equilibrium.
THEOREM 3 (Routh-Hurwitz Conditions)
Let 𝐽 = (
𝑓𝑥 (𝑥∗ , 𝑦∗ )
𝑔𝑥 (𝑥∗ , 𝑦∗ )
𝑓𝑦 (𝑥∗ , 𝑦∗ )
)
𝑔𝑦 (𝑥∗ , 𝑦∗ )
be the Jacobian matrix of the non – linear system
𝑑𝑥
= 𝑓(𝑥, 𝑦)
𝑑𝑡
𝑑𝑦
= 𝑔(𝑥, 𝑦)
𝑑𝑡
evaluate at the critical point (𝑥∗ , 𝑦∗ ). Then the critical point (𝑥∗ , 𝑦∗ )
I.
II.
III.
is asymptotically stable if 𝑇𝑟𝑎𝑐𝑒(𝐽) < 0 and 𝐷𝑒𝑡 (𝐽) > 0
is stable but not asymptotically stable if 𝑇𝑟𝑎𝑐𝑒 = 0 and 𝐷𝑒𝑡 (𝐽) > 0
is unstable if 𝑇𝑟𝑎𝑐𝑒(𝐽) < 0 and 𝐷𝑒𝑡 < 0
7
We shall now apply the Routh-Hurwitz conditions in studying the stability of the endemic equilibrium. The
Jacobian matrix associated with the system (3.1) is
∗
∗
𝐽(𝑆 , 𝐼 , 𝑅
∗)
−𝛽𝐼 ∗ − 𝜇
= [ 𝛽𝐼 ∗
0
𝛽𝑆 ∗
𝛽𝑆 − (𝜇 + 𝛿 + 𝛼)
𝛼
∗
0
0]
−𝜇
Clearly, −𝜇 is an Eigen value, the other two Eigen values are negative if the Routh – Hurwitz conditions hold.
That is,
I.
II.
Trace of 𝐺 < 0
Determinant of 𝐺 > 0
Where
−𝛽𝐼 ∗ − 𝜇
𝐺=[
𝛽𝐼 ∗
𝛽𝑆 ∗
]
𝛽𝑆 − (𝜇 + 𝛿 + 𝛼)
∗
The trace of G is given as
𝑇𝑟𝑎𝑐𝑒 𝐺 = −𝛽𝐼 ∗ − 𝜇 + 𝛽𝑆 ∗ − (𝜇 + 𝛿 + 𝛼)
= −𝛽(𝑅0 − 1)
(𝛿 + 𝛼 + 𝜇)
𝜇
−𝜇+𝛽
− (𝜇 + 𝛿 + 𝛼)
𝛽
𝛽
= −𝛽(𝑅0 − 1)
𝜇
−𝜇
𝛽
< 0 for 𝑅0 > 1
The Determinant of G is given as
𝐷𝑒𝑡 𝐺 = −(𝛽𝐼 ∗ + 𝜇)[𝛽𝑆 ∗ − (𝜇 + 𝛿 + 𝛼)] − 𝛽 2 𝑆 ∗ 𝐼 ∗
= − (𝛽(𝑅0 − 1)
(𝛿 + 𝛼 + 𝜇)
(𝛿 + 𝛼 + 𝜇)
𝜇
𝜇
(𝑅0 − 1) )
+ 𝜇) [𝛽
− (𝜇 + 𝛿 + 𝛼)] − (𝛽 2
𝛽
𝛽
𝛽
𝛽
= − (𝛽(𝑅0 − 1)
𝜇
+ 𝜇) [0] − ((𝛿 + 𝛼 + 𝜇)(𝑅0 − 1)𝜇)
𝛽
= −((𝛿 + 𝛼 + 𝜇)(𝑅0 − 1)𝜇)
< 0 for 𝑅0 > 1
Therefore one of the conditions given above for negativity of the Eigen values is violated. This implies that the
endemic equilibrium of the model system 3.1 is unstable for 𝑅0 > 1.
6.0 MODEL SIMULATION
We now perform a numerical simulation of the model using parameter values as listed in the table below
8
Parameter
Value
𝑄0
3
𝛽
[0.003, 0.03]
𝛼
0.4
𝛿
0.0023
𝜇
0.02
Table: Parameter values used in simulation
Susceptibles
1000
500
0
0
2
4
6
8
10 12
Time
14
16
18
20
Infectives
1000
500
0
Recovered
Beta = 0.03
0
2
4
6
8
10 12
Time
14
16
18
1000
500
0
0
2
4
6
8
10 12
Time
14
16
18
1000
500
0
20
Figure 1: Simulation of the mass action model with
𝛽 = 0.003
0
2
4
6
8
10 12
Time
14
16
18
20
0
2
4
6
8
10 12
Time
14
16
18
20
0
2
4
6
8
10 12
Time
14
16
18
20
1000
500
0
20
Recovered
Infectives
Susceptibles
Beta = 0.003
1000
500
0
Figure 2: Simulation of the mass action model with
𝛽 = 0.03
9
7.0 CONCLUSION
We analyzed a mass action model with vital dynamics. Most of our result depends on the behavior of 𝑅0 . For
example if 𝑅0 < 1 then the disease free equilibrium is both locally and globally asymptotically
stable. Also the endemic equilibrium exist only if 𝑅0 > 1. This implies that control measures should
enforce those policies that reduces 𝑅0 below unity since the disease can be earidicated from the
population if 𝑅0 is forced below unity. The parameter that can be easily manipulated to bring 𝑅0
below unity is the contact rate. As seen from figures 1 and 2, increasing the contact rate have
negative effect on the transmission dynamics of the disease as it results in tremendous increase in
the number of infectives causing the disease to take over the entire population in a short period of
time.
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