Problem 8 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Anderson and May \(^{4}\) give the following model of immune effector cells (helper and cytotoxic T-cells), \(E,\) that limit viral population, \(V\), growth in a human body. $$ \begin{array}{l} d E / d t=\Lambda-\mu E+\epsilon V E \\ d V / d t=r V-\sigma V E \end{array} $$ a. \(\Lambda\) is intrinsic production rate of effector cells from bone marrow. Give similar meaning to each of the other four terms on the RHS of Equations 18.64 . b. Find the equilibrium effector cell population, \(\hat{E},\) in the absence of virus. c. Suppose an inoculum \(V_{0}\) of virus is introduced into the body with \(E=\hat{E}\). Find conditions on \(r\), \(\sigma,\) and \(\hat{E}\) in order that the viral population will increase. d. The Jacobian matrix at any \((E, V)\) is $$ J(E, V)=\left[\begin{array}{rc} -\mu+\epsilon V & \epsilon E \\ -\sigma V & r-\sigma V \end{array}\right] $$ Show that $$ J(\hat{E}, 0)=\left[\begin{array}{rr} -\mu & \sigma \frac{\Lambda}{\mu} \\ 0 & r-\sigma \frac{\Lambda}{\mu} \end{array}\right] $$ e. The characteristic values of the upper diagonal matrix \(J(\hat{E}, 0)\) are the diagonal entries, \(-\mu\) and \(r-\sigma \frac{\Lambda}{\mu} .\) What happens to a small introduction of virus into a healthy individual if both characteristic values are negative? f. In order that the viral population to expand it is necessary that \(r-\sigma \frac{\Lambda}{\mu}>0 .\) What is the role of \(r\) in the model? g. If that condition is met and the viral population increases, show that there will be an equilibrium state, $$ E^{*}=r / \sigma, \quad V^{*}=\frac{\mu r-\Lambda \sigma}{\epsilon r} $$ It is clear that in order for \(V^{*}\) to be positive, we must have (again) $$ \mu r-\Lambda \sigma>0 \quad \text { so that } \quad r>\frac{\Lambda \sigma}{\mu} $$ h. Anderson and May report this system to be asymptotically stable, at \(\left(E^{*}, V^{*}\right),\) but only weakly so, meaning that it is subject to wide oscillations. Show that $$ J\left(E^{*}, V^{*}\right)=\left[\begin{array}{cc} -\frac{\Lambda \sigma}{r} & \epsilon \frac{r}{\sigma} \\ -\sigma \frac{\mu r-\Lambda \sigma}{\epsilon r} & 0 \end{array}\right] $$ i. The characteristic equation of a \(2 \times 2\) matrix \(M\) is is \(s^{2}-\operatorname{trace}(M) s+\operatorname{det}(M)=0\). Show that the characteristic equation of \(J\left(E^{*}, V^{*}\right)\) is $$ s^{2}+\frac{\Lambda \sigma}{r} s+\mu r-\Lambda \sigma=0 $$ j. The roots of the characteristic equation 18.65 are $$ \frac{-\frac{\Lambda \sigma}{r} \pm \sqrt{\left(\frac{\Lambda \sigma}{r}\right)^{2}-4(\mu r-\Lambda \sigma)}}{2} $$ Argue that the real part is negative so that the system is stable. Note: Two cases: $$ \left(\frac{\Lambda \sigma}{r}\right)^{2}-4(\mu r-\Lambda \sigma)>0 \quad \text { and } \quad\left(\frac{\Lambda \sigma}{r}\right)^{2}-4(\mu r-\Lambda \sigma)<0 $$ k. Use \(\Lambda=1, \mu=0.5, \epsilon=0.02, r=0.25,\) and \(\sigma=0.01\) and compute \(E^{*}, V^{*},\) and the stability at \(\left(E^{*}, V^{*}\right)\) l. Let \(E_{0}=2, V_{0}=1, \Lambda=1, \mu=0.5, \epsilon=0.02, r=0.25,\) and \(\sigma=0.01 .\) Approximate the solutions to Equations 18.64 using the trapezoid rule. Observe that the rise in viral load precedes the increase in effector cells.

Step-by-Step Solution

Answer

The viral population increases when \( r > \sigma \frac{\Lambda}{\mu} \). At equilibrium, \( E^* = \frac{r}{\sigma} \) and \( V^* = \frac{\mu r - \Lambda \sigma}{\epsilon r} \). System stability is ensured by negative real parts of characteristic equation roots.

Step 1: Understanding Terms in Differential Equations

The terms in the immune effector cell and viral population model represent different processes. \( \mu E \) represents the natural death rate of effector cells. \( \epsilon V E \) is the stimulation effect of the virus load on effector cell proliferation. \( r V \) stands for the natural replication rate of the virus, while \( \sigma V E \) indicates the rate at which effector cells inhibit virus replication.

Step 2: Equilibrium Effector Cell Population Without Virus

In the absence of the virus \((V = 0)\), the differential equation for \(E\) simplifies to \( \frac{dE}{dt} = \Lambda - \mu E \). Setting \( \frac{dE}{dt} = 0 \) for equilibrium, we solve \( \Lambda - \mu \hat{E} = 0 \) to find \( \hat{E} = \frac{\Lambda}{\mu} \).

Step 3: Conditions for Viral Population Increase

When the virus is introduced, the condition for virus growth is derived from \( \frac{dV}{dt} > 0 \), leading to \( rV - \sigma V \hat{E} > 0 \). Simplifying, we find \( r > \sigma \hat{E} = \sigma \frac{\Lambda}{\mu} \).

Step 4: Jacobian at \\(\hat{E}, 0\\)

To show that \( J(\hat{E}, 0) = \begin{bmatrix} -\mu & \sigma \frac{\Lambda}{\mu} \ 0 & r - \sigma \frac{\Lambda}{\mu} \end{bmatrix} \), substitute \( \hat{E} = \frac{\Lambda}{\mu} \) and \( V = 0 \) into the Jacobian matrix \( J(E, V) \).

Step 5: Interpretation of Negative Characteristic Values

When both \(-\mu\) and \(r - \sigma \frac{\Lambda}{\mu}\) are negative, any introduction of virus will not grow and the system is stable because \(E\) and \(V\) will return to equilibrium.

Step 6: Role of r in Viral Population Dynamics

The parameter \(r\) represents the natural reproductive rate of the virus. A higher \(r\) increases the potential for viral growth unless controlled by \(\sigma \hat{E}\).

Step 7: Equilibrium States and Their Stability

When \(r - \sigma \frac{\Lambda}{\mu} > 0\), the viral population expands to equilibrium at \(E^{*} = \frac{r}{\sigma}\) and \(V^{*} = \frac{\mu r - \Lambda \sigma}{\epsilon r} \). To ensure \(V^{*} > 0\), \(r > \frac{\Lambda \sigma}{\mu}\) is needed.

Step 8: Jacobian at Equilibrium \\(E^*, V^*\\)

At \((E^*, V^*)\), the Jacobian \( J(E^*, V^*) \) is \( \begin{bmatrix} -\frac{\Lambda \sigma}{r} & \epsilon \frac{r}{\sigma} \ -\sigma \frac{\mu r - \Lambda \sigma}{\epsilon r} & 0 \end{bmatrix} \). Substituting known values verifies this form.

Step 9: Characteristic Equation of the Jacobian

The characteristic equation for \(2 \times 2\) matrix \(J(E^*, V^*)\) is derived as \(s^2 + \frac{\Lambda \sigma}{r} s + \mu r - \Lambda \sigma = 0\).

Step 10: Stability Analysis Using Characteristic Equation Roots

The roots of \(s^2 + \frac{\Lambda \sigma}{r} s + \mu r - \Lambda \sigma = 0\) have negative real parts if it has complex roots or both roots are negative real, ensuring system stability.

Step 11: Calculating Equilibrium and Stability with Given Values

Substituting \(\Lambda=1, \mu=0.5, \epsilon=0.02, r=0.25, \sigma=0.01\) gives \(E^* = 25\), \(V^* = 1.5\), and the system is stable as verified by the Jacobian matrix.

Step 12: Approximating Solutions with Trapezoid Rule

Use initial values \(E_0=2, V_0=1, \Lambda=1, \mu=0.5, \epsilon=0.02, r=0.25, \sigma=0.01\) to approximate solutions to the equations using the trapezoid rule, noting that viral load initially rises before effector cells respond.

Key Concepts

Equilibrium Solutions

In the context of differential equations in biology, equilibrium solutions refer to a state where the variables remain constant over time. For the immune effector cells and viral population model outlined by Anderson and May, equilibrium solutions occur when the rate of change for both variables, the effector cells \(E\) and the viral population \(V\), is zero. This implies that the biological system has reached a stable state where the internal dynamics no longer produce changes in the population levels.

To find the equilibrium state for effector cells in the absence of the virus, we start by simplifying the equation \(\frac{dE}{dt} = \Lambda - \mu E\) for \(V = 0\). Setting \(\frac{dE}{dt} = 0\), we solve for \(E\), leading to \(\hat{E} = \frac{\Lambda}{\mu}\). This result indicates that the equilibrium population of effector cells is directly influenced by the intrinsic production rate \(\Lambda\) and their natural death rate \(\mu\). When viruses are present, similar calculations are required to ensure stability or growth conditions, focusing on the equilibrium of both \(E\) and \(V\).

Stability Analysis

Stability analysis in the realm of biology and differential equations is crucial to understand the behavior of a system near its equilibrium. For the model described, stability analysis helps predict whether a small introduction of the virus to a previously stable state will cause the system to return to equilibrium or diverge.

In this model, stability is interpreted through the characteristic values of the Jacobian matrix, an approach discussed in later sections. These characteristic values, also known as eigenvalues, indicate the system's tendency to return to equilibrium or become unstable. Negative characteristic values suggest that perturbations (such as a sudden increase in viral load) will decay over time, resulting in stability. However, if these values are positive, it implies that the system could be driven further away from equilibrium, causing instability and potential outbreaks of the viral population.

It's essential to calculate these values accurately and consider both environmental and biological variables, such as natural reproduction rates and immune response factors, to make precise predictions about the system's behavior.

Jacobian Matrix

The Jacobian matrix is a powerful tool used in the analysis of dynamic systems to determine their behavior near equilibrium points. In this context, the matrix consists of partial derivatives which represent how changes in one variable affect the other. This matrix helps in understanding the interactions between effector cells and viruses in the human body.

For Anderson and May's model, the Jacobian matrix at any point \( (E, V) \) is given by: \J(E, V) = \begin{bmatrix} -\mu + \epsilon V & \epsilon E \ -\sigma V & r - \sigma E \end{bmatrix}.\By substituting the equilibrium values for \(E\) and \(V\), this general matrix simplifies, offering insights into the local stability of the dynamic system at specific points.

Computing the Jacobian at equilibrium allows us to find characteristic values, which tell us if the point is stable or not. A crucial part of stability analysis is solving for these eigenvalues. If both are negative or have a negative real part, the system is stable at that point, and any perturbation will fade over time. Understanding and building the Jacobian matrix is a core skill in applying mathematical models to biological systems, providing insights into how shifts in population levels can impact the entire system.

Viral Dynamics Model

The viral dynamics model explored in this exercise represents the interaction between immune effector cells and a viral population, using differential equations to simulate biological processes within a host. Here, elements like production and death rates, as well as the interaction between cells and viruses, are mathematically defined, capturing the complex dynamics of viral infections.

This model postulates that the effector cells, which are part of the immune response, actively limit the growth of the virus population through their interactions. The production rate of effector cells is denoted by \(\Lambda\), and they naturally die at a rate of \(\mu E\). Likewise, viruses replicate at a rate \(rV\), but their numbers are kept in check by effector cells at a rate \(\sigma V E\). Understanding these terms within the equations helps in predicting viral behavior, especially in cases of infection introduction to healthy individuals.

Analyzing this model sheds light on critical thresholds needed to prevent viral outbreaks, such as conditions where viral dynamics allow population increase versus immune control. The balance between growth rates of the virus and immune response can be decisive in predicting health outcomes, making this model a practical tool in epidemiology and immunology. The viral dynamics model is thus pivotal in formulating effective response strategies in healthcare and research.