Coinfection models

Introduction

• Pathogens at times interact.
• Interaction can be direct or – more common – immune system mediated.
• Important interactions among pathogens:
  • HIV + TB (and HIV with others)
  • flu + bacteria
  • helminths + malaria/allergies
• Coinfection models range from relatively simple to very complex.
• There is nothing fundamentally different between these and the models we have seen so far.

Coinfection models

• Models have been used to study coinfection.
• The models range from relatively simple to very complex.
• We’ll briefly look at a few examples.
• There is nothing fundamentally different between these and the models we have seen so far.

Notes:

• Most model equations in this slide-deck were generated by ChatGPT based on the original papers and not checked for accuracy.
• Examples were chosen purely based on model suitability.

TB/HIV model example

“In‑vivo mathematical study of co‑infection dynamics of HIV‑1 and M tuberculosis”, Magombedze et al 2008, JBS.

Question: How do HIV and TB influence each other to impact infection outcomes.

Approach: Combined TB and HIV model, explored to understand how interactions among pathogens and the immune response influence infection outcomes.

TB/HIV model example - TB model

\[ \begin{align} \dot{M}_r &= s_m + p_m\,T_b\,M_r - k_i\,T_b\,M_r - \mu_m\,M_r && \text{resting macrophages} \\ \dot{M}_{ib} &= k_i\,T_b\,M_r - k_b\,M_{ib} - k_a\,\frac{M_{ib}}{M_{ib}+A}\,T - k_l\,M_{ib}\,C_b - \mu_{mi}\,M_{ib} && \text{infected macrophages} \qquad \\ \dot{T}_b &= k_b\,N_b\,M_{ib} - \gamma_1\,T_b\,M_r - \gamma_2\,T_b\,C_b + r_m\,T_b + k_l\,N_c\,M_{ib}\,C_b && \text{extracellular Mtb} \\ \dot{T} &= s_t + r_1\,\frac{T\,T_b}{T_b + B} - \mu_t\,T && \text{CD4$^{+}$T cells} \\[4pt] \dot{C}_b &= s_b + p_b\,\frac{M_{ib}}{M_{ib} + R_b}\,T\,C_b - \mu_{cb}\,C_b && \text{Mtb‑specific CTL} \end{align} \]

TB/HIV model example - HIV model

\[ \begin{align} \dot{T} &= s_t + r\,\frac{T\,V}{V + A_a} - \beta\,\frac{V\,T}{1 + a_1\,C_{hv}} - r_2\,\frac{T\,V}{T + R} - \mu_t\,T && \text{CD4$^+$ T cells} \\ \dot{T}^* &= \beta\,\frac{V\,T}{1 + a_1\,C_{hv}} - h_v\,T^* C_{hv} - \alpha_t\,T^* && \text{HIV-infected CD4$^+$ T cells} \\[6pt] \dot{C}_{hv} &= s_v + p_v\,V\,T\,C_{hv} - \mu_{cv}\,C_{hv} && \text{HIV-specific CTLs} \\[6pt] \dot{V} &= \frac{N_v\,\alpha_t\,T^*}{1 + a_2\,C_{hv}} - \mu_v\,V && \text{free HIV virions} \end{align} \]

TB/HIV model example - combined model

TB/HIV model example

Time course of model variables during co-infection. ChatGPT apologizes for this sub-par job, it tried hard.

Influenza and bacteria example

“Quantifying the therapeutic requirements and potential for combination therapy to prevent bacterial coinfection during influenza” by Amber Smith, 2017 JPP

Question: What is the impact of antiviral or antibacterial drugs during influenza-bacteria co-infections?

Approach: Exploration of several fairly simple mathematical models, comparison with data.

Influenza and bacteria example

Model Schematic

Influenza and bacteria example

\[ \begin{align} \dot{T} &= -\beta\,T\,V && \text{(susceptible target cells \(T\))} \\[4pt] \dot{I}_1 &= \beta\,T\,V - k\,I_1 - \mu\,P\,I_1 && \text{(eclipse‑phase infected cells \(I_1\))} \\[4pt] \dot{I}_2 &= k\,I_1 - \delta I_2 - \mu\,P\,I_2 && \text{(virus‑producing infected cells \(I_2\))} \\[4pt] \dot{V} &= p\,I_2 - c\,V && \text{(free virus \(V\))} \\[4pt] \end{align} \]

Influenza and bacteria example

\[ \begin{align} \dot{T} &= -\beta\,T\,V && \text{(susceptible target cells \(T\))} \\[4pt] \dot{I}_1 &= \beta\,T\,V - k\,I_1 - \mu\,P\,I_1 && \text{(eclipse‑phase infected cells \(I_1\))} \\[4pt] \dot{I}_2 &= k\,I_1 - \delta I_2 - \mu\,P\,I_2 && \text{(virus‑producing infected cells \(I_2\))} \\[4pt] \dot{V} &= (1+\color{red}{aP^z})p I_2 - c\,V && \text{(free virus \(V\))} \\[4pt] \dot{P} &= r\,P\!\left(1 - \frac{P}{K_P\!\bigl(1 + \psi V\bigr)}\right) \\ & - \gamma_{M_A} \frac{n^x M_A}{P^x+n^x M_A} M_A P\bigl(1 - \frac{\phi V}{(K_{PV}+V)}\bigr) && \text{(pneumococcal bacteria \(P\))} \end{align} \] \(M_A\) are macrophages, not explicitly modeled, kept fixed.

Influenza and bacteria example

\[ \begin{align} \dot{T} &= -\beta\,T\,V && \text{(susceptible target cells \(T\))} \\[4pt] \dot{I}_1 &= \beta\,T\,V - k\,I_1 - \mu\,P\,I_1 && \text{(eclipse‑phase infected cells \(I_1\))} \\[4pt] \dot{I}_2 &= k\,I_1 - \delta I_2 - \mu\,P\,I_2 && \text{(virus‑producing infected cells \(I_2\))} \\[4pt] \dot{V} &= \color{red}{(1-\epsilon_v)} (1+aP^z) p I_2 - c\,V && \text{(free virus \(V\))} \\[4pt] \dot{P} &= r\,P\!\left(1 - \frac{P}{K_P\!\bigl(1 + \psi V\bigr)}\right) \\ & - \gamma_{M_A} \frac{n^x M_A}{P^x+n^x M_A} M_A P\bigl(1 - \frac{\phi V}{(K_{PV}+V)}\bigr) -\color{red}{\epsilon_a P} && \text{(pneumococcal bacteria \(P\))} \end{align} \] More alternatives discussed in the paper.

Influenza and bacteria example

Exploring the impact of treatment.

Summary

• Coinfection models are generally a bit larger since they need to account for both pathogens.
• One needs to know (or assume) the underlying interaction processes.
• Models are most useful at potentially discriminating between different possible mechanisms.