Unit - 4

Equilibrium points

Q1) Give the representation as a System of First Order Equations.

A1)

For the second order linear ordinary differential equation-

We can write this as system of two first order equations

In the case of mass-spring oscillator, v is the velocity. In the case of a circuit in series, we typically write the equation in terms of the charge q,

We can write this as the system

In this case I is the current.

Q2) What are equilibrium points?

A2)

As in the of one equation, a point x0, y0 satisfying

f(x0, y0) = 0 and g(x0, y0) = 0

Called a critical point, or equilibrium point, of the autonomous system (2) The corresponding constant solution (x(t), y(t)) = (x0, y0) is called an equilibrium solution.

The set of all critical points is called the critical point set.

Example: for

The direction field,

Gives vectors with positive slope in quadrants I and III and negative slope in quadrants I and IV

Q3) Explain predatory-prey model?

A3)

Predatory-prey equation is also known as the Lotka-Volterra equations.

The first order non-linear differential equations used to describe the dynamics of biological systems, one as predator and other is prey.

The populations change is represented as-

Where

x is the number of prey

y is the number of predator

instantaneous growth rates of the two populations;

t = time

are the parameters.

Q4) Give the physical meaning for predatory-prey equations.

A4)

Predatory-prey model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations

1. The prey population finds ample food at all times.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of change of population is proportional to its size.
4. During the process, the environment does not change in favour of one species, and genetic adaptation is inconsequential.
5. Predators have limitless appetite.

Q5) What is population equilibrium?

A5)

Population equilibrium occurs in the model when neither of the population levels is changing, which means when both of the derivatives are equal to 0:

The above system of equations yields two solutions:

{x = 0, y = 0}

And

Q6) Explain epidemic model of influenza.

A6)

Every year, influenza causes high morbidity and mortality especially among the immune-compromised persons worldwide. The emergence of drug resistance has been a major challenge in curbing the spread of influenza.

Influenza is a contagious respiratory illness caused by influenza viruses. There are three major types of flu viruses: types A, B, and C. The majority of human infections are caused by types A and B. Of major concern is influenza A virus which is clinically the most vicious. It is a negative-sense single-stranded RNA virus with eight gene segments. The segmented nature of influenza A virus genome allows the exchange of gene segments between viruses that coinfect the same cell

This process of genetic exchange is termed reassortment. Reassortment leads to sudden changes in viral genetics and to susceptibility in hosts. Influenza A virus has a wide range of susceptible avian hosts and mammalian hosts such as humans, pigs, horses, seals, and mink. In addition, the virus is able to repeatedly switch hosts to infect multiple avian and mammalian species. The unpredictability of influenza A virus evolution and interspecies movement creates continual public health challenges.

Influenza can be prevented by getting vaccination each year. However, given that the virus mutates rapidly, a vaccine made for one year may not be useful in the following year. In addition, antigenic drift in the virus may occur after the year’s vaccine has been formulated, rendering the vaccine less protective, and hence, outbreaks can easily occur especially among high-risk individuals. According to, other preventive actions include staying away from people who are sick, covering coughs and sneezes, and frequent handwashing.

Q7) What is the mathematical formulation of epidemic model of influenza?

A7)

The model subdivides the total population into five compartments: Susceptible (S), Vaccinated (V), Infected with Wild-type strain (Iw) Infected with Resistant strain (IR), and Recovered (R). Individuals in a given compartment are assumed to have similar characteristics. Parameters vary from compartment to compartment but are identical for all individuals in a given compartment. Individuals enter the population at the rate of π, and all recruited individuals are assumed to be susceptible. The Susceptible get infected after effective contact with either the Infected with Wild-type strain or the Infected with Resistant strain. The force of infection is given by either (Infection by Wild-type strain) or   (Infection by Resistant strain), where

Parameters  and  refer to the transmission rate of wild-type strain and resistant strain, respectively. Parameter b is the rate of developing drug resistance. The susceptible can only be infected by one strain at a time. The rate of vaccination is ϕ. The vaccinated can also become infected with either the wild-type strain or the resistant strain. This depends on the vaccine efficacy. When the vaccine efficacy is 100%, the vaccinated cannot become infected. Individuals who are infected with the wild-type strain are treated and recover at the rate of α, while those who are infected with the resistant strain recover at the rate of  . The wild-type strain is assumed to mutate to resistant strain, and hence, those infected with the wild type join those infected with the resistant strain at the rate of b. Individuals with wild-type strain and those with resistant strain suffer disease-induced death at the rates   and , respectively. The recovered lose immunity at the rate of ϑ joins the susceptible class. Individuals in all the epidemiological compartments suffer natural death at the rate of μ. The model diagram is given in Figure-

Q8) Given that the initial conditions of system (1) are S(0) > 0, V(0) 0, and the solutions S(t), V(t),   and R(t) are non-negative for all t > 0.

A8)

Assume that

Thus  , and it follows directly from the first equation of system (1) that

……… (2)

Using the integrating factor method to solve inequality (2), we have

………(3)

Integrating both sides, we get

…………(4)

Where C is the constant. Hence,

S(0)……….. (5)

Hence, )>0

From the second equation in system (1), we obtain

………. (6)

Hence

……… (7)

Similarly, it can be shown that

…….. (8)

Therefore, all the solutions of system (1) with nonnegative initial conditions will remain nonnegative for all time t > 0.

Q9) Give the expression invariant Region.

A9)

We show that the total population is bounded for all time t > 0. The analysis of system (1) will therefore be analyzed in a region  of biological interest. Thus, we have the following theorem on the region that system (1) is restricted to.

Theorem:

The feasible region  defined by

…….. (i)

The initial conditions is positively invariant and attracting with respect to system (1) for all t > 0.

Proof. Summing up the equations in (1) [above], we obtain that the total population satisfies the following differential equation:

……. (ii)

In the absence of influenza infection, it follows that

…….. (iii)

It can easily be seen that

…….. (iv)

From (iv), we observe that as t , , so if then

On other hand if then N will decrease to This means that therefore N(t) is bounded above. Subsequently are bounded above. Thus, in  system (1) is well posed. Hence, it is sufficient to study the dynamics of the system in 