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

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

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.