Viral Infection models

2023-07-07

Basic acute virus infection model

\[ \begin{aligned} \textrm{Uninfected Cells} \qquad \dot{U} & = ? \\ \textrm{Infected Cells} \qquad \dot{I} & = ? \\ \textrm{Virus} \qquad \dot{V} & = ? \end{aligned} \]

Basic acute virus infection model

\[ \begin{aligned} \textrm{Uninfected Cells} \qquad \dot{U} & = - bUV \\ \textrm{Infected Cells} \qquad \dot{I} & = bUV - d_I I \\ \textrm{Virus} \qquad \dot{V} & = pI - d_V V - bg UV \end{aligned} \]

Basic chronic virus infection model

\[ \begin{aligned} \textrm{Uninfected Cells} \qquad \dot{U} & = \color{blue}{n -d_U U} - bUV \\ \textrm{Infected Cells} \qquad \dot{I} & = bUV - d_I I \\ \textrm{Virus} \qquad \dot{V} & = pI - d_V V - bg UV \end{aligned} \]

Basic virus infection models

Dynamics of simple viral infection models. A) acute infection. B) chronic infection.

Notation variability

I’m trying to use the same letters for variables and parameters in these materials.
If you read the literature, you’ll see all kinds of variants.

\[ \dot{T} = s - uT - \beta T V \\ \dot{T^*} = \beta T V - d T^* \\ \dot{V} = NdT^* - c V - \beta g TV \] \[ \dot{x} = \lambda - dx - \beta x v \\ \dot{y} = \beta x v - a y \\ \dot{v} = \kappa y - u v - \beta g xv \]

Diagrams and Models

It is important to go back and forth between words, diagrams, equations.
Diagrams specify a model somewhat, but not completely.

Diagrams and Models

The virus model diagram could be implemented as continuous or discrete (or stochastic) model.

\[ \begin{aligned} \dot{U} & = n -d_U U - bUV \\ \dot{I} & = bUV - d_I I \\ \dot{V} & = pI - d_V V - bg UV \end{aligned} \]

\[ \begin{aligned} U_{t+dt} & = U_{t} + dt(n -d_U U_t - bU_tV_t) \\ I_{t+dt} & = I_{t} + dt( bU_tV_t - d_I I_t )\\ V_{t+dt} & = V_{t} + dt( pI_t - d_V V_t - bg U_t V_t ) \end{aligned} \]

Model Exploration

Explore the “Basic Virus Model” app in DSAIRM

Virus and Immune Response Models

The immune response is incredibly complex, we still don’t know how to model it in much detail.
We can nevertheless build and explore models that are a (hopefully) good balance between realism and abstraction.

Virus and Immune Response Model 1

U - uninfected cells
I - infected cells
V - (free) virus
F - innate (interferon) response
T - adaptive (T-cell) response

Virus and Immune Response Model 1

\[\begin{aligned} \dot U & = - b U V \\ \dot I & = b U V - d_I I - k_T T I \\ \dot V & = \frac{p}{1+k_F F}I - d_V V - gb UV \\ \dot F & = r_F I - d_F F \\ \dot T & = r_T T F - d_T T \end{aligned}\]

Virus and Immune Response Model 2

U - uninfected cells
I - infected cells
V - (free) virus
F - innate (interferon) response
T - adaptive (T-cell) response

Virus and Immune Response Model 2

\[\begin{aligned} \dot U & = n - d_U U - b U V \\ \dot I & = bUV - d_I I - k_T T I \\ \dot V & = \frac{p}{1+k_F F}I - d_V V - gb UV \\ \dot F & = r_F I - d_F F \\ \dot T & = r_T T V - d_T T \end{aligned}\]

Models 1 and 2

\[\begin{aligned} \dot U & = - b U V \\ \dot I & = b U V - d_I I - k_T T I \\ \dot V & = \frac{p}{1+k_F F}I - d_V V - gb UV \\ \dot F & = r_F I - d_F F \\ \dot T & = r_T T F - d_T T \end{aligned}\]

\[\begin{aligned} \dot U & = \color{blue}{n - d_U U} - b U V \\ \dot I & = bUV - d_I I - k_T T I \\ \dot V & = \frac{p}{1+k_F F}I - d_V V - gb UV \\ \dot F & = r_F I - d_F F \\ \dot T & = \color{blue}{r_T T V} - d_T T \end{aligned}\]

Virus and Immune Response Model 3

U - uninfected cells
I - infected cells
V - (free) virus
F - innate immune response
T - CD8 T-cells
B - B-cells
A - Antibodies

Model Diagram

Model Equations

\[ \dot U = n - d_U U - bUV\\ \dot I = bUV - d_I I - k_T T I\\ \dot V = \frac{pI}{1+s_F F} - d_V V - b UV - k_A AV \\ \dot F = p_F - d_F F + \frac{V}{V+ h_V}g_F(F_{max}-F) \\ \dot T = F V g_T + r_T T\\ \dot B = \frac{F V}{F V + h_F} g_B B \\ \dot A = r_A B - d_A A - k_A A V \]

Model Exploration

Explore any/all of the “IR” models in the virus models section

Homework

If you haven’t already, take a look at the “Basic Bacteria Model” app and go through - in as much detail as you have time - the “What to do” tasks. You can try yourself first, or look at the solution right away.
Take an especially close look at the solution for task 6, we will use a similar approach tomorrow.