Macroparasite Infection models

Andreas Handel

2025-07-18

Introduction

  • Macroparasites are “big things” (> bacteria)
  • Mostly extracellular
  • Important for human health:
    • Malaria
    • Worms/Helminths

Parasite Infection models

  • Models have been used to study parasite infections, though less than for common viruses and bacteria.
  • 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.

Malaria Example 1

“On the Control of Acute Rodent Malaria Infections by Innate Immunity” by Kochin et al (2010) PLoS One.

Question: Can differences in innate immunity explain infection patterns of different malaria strains?

Approach: Simple mathematical model fit to data from rodent malaria co-infections with two malaria strains (AJ and AS). Quality of fit is used to discriminate between model variants/hypotheses.

Malaria Example 1

\[ \begin{aligned} \dot{P} & = r P - k P I \qquad & \textrm{Pathogen}\\ \dot{I} & = \alpha P (j-I) - d I \qquad & \textrm{Innate IR} \end{aligned} \]

Kochin et al (2010) PLoS One

Malaria Example 1

\[ \begin{aligned} \dot{P_{AJ}} & = r_{AJ} P_{AJ} - k_{AJ}P_{AJ} I \qquad & \textrm{AJ strain}\\ \dot{P_{AS}} & = r_{AS} P_{AS} - k_{AS}P_{AS} I \qquad & \textrm{AS strain}\\ \dot{I} & = (\alpha_{AS} P_{AS} + \alpha_{AJ} P_{AJ}) (j-I) - d I \qquad & \textrm{Innate IR} \end{aligned} \]

  • Simple model
  • Ignores intracellular processes
  • Only a single equation for innate immune response

Malaria Example 1

Experimental data and models for the dynamics of strains AS (red dashed lines) and AJ (blue solid lines). (I) AJ has a higher growth rate than AS; (II) AJ induces innate immunity more slowly than AS; or (III) AJ is less susceptible to killing by innate immunity than AS. Kochin et al (2010) PLoS One.

Malaria Example 2

“Host Control of Malaria Infections: Constraints on Immune and Erythropoeitic Response Kinetics” by McQueen & McKenzie (2008) PLoS Comp Bio.

Question: Which components of the immune response help to control malaria infections?

Approach: Complex mathematical model explored for different parameter values.

Malaria Example 2

Model Schematic

Malaria Example 2

\[ \begin{align} \dot{I}_1 &= f\,m\,V - \Bigl(k_I + \sum_m \mathrm{Att}_m\,j_{m,1}\Bigr) I_1 && \text{(young infected‑RBC compartment \(I_1\))} \\[4pt] \dot{I}_n &= k_I I_{n-1} - \Bigl(k_I + \sum_m \mathrm{Att}_m\,j_{m,n}\Bigr) I_n && \text{(infected‑RBC compartment \(I_n\), $2\le n\le N_{cI}+1$)} \\[8pt] \dot{m} &= p\,k_I I_{N_{cI}} - m\!\left(f\,V + \tfrac{1}{T_{Dm}}\right) - m\sum_m \mathrm{Att}_m\,j_{m,m} + L(t) && \text{(merozoite density \(m\))} \\[8pt] \dot{R}_1 &= E_S(t) - k_R R_1 - f\,m\,R_1 && \text{(first reticulocyte compartment \(R_1\))} \\[4pt] \dot{R}_n &= k_R R_{n-1} - (k_R + f\,m) R_n && \text{(reticulocyte compartment \(R_n\), $2\le n\le N_{cR}+1$)} \\[8pt] \dot{M}_1 &= k_R R_{N_{cR}} - k_M M_1 - f\,m\,M_1 && \text{(first mature‑RBC compartment \(M_1\))} \\[4pt] \dot{M}_n &= k_M M_{n-1} - (k_M + f\,m) M_n && \text{(mature‑RBC compartment \(M_n\), $2\le n\le N_{cM}+1$)} \\[8pt] \dot{S}_1 &= k_M M_{N_{cM}} - k_S S_1 - f\,m\,S_1 && \text{(first senescent‑RBC compartment \(S_1\))} \\[4pt] \dot{S}_n &= k_S S_{n-1} - (k_S + f\,m) S_n && \text{(senescent‑RBC compartment \(S_n\), $2\le n\le N_{cS}+1$)} \\[8pt] \dot{E}_S &= \begin{cases} \lambda_{ES}\,(W - E_S), & E_{SMN} < W < E_{SMX} \\[4pt] \lambda_{ES}\,(E_{SMX} - E_S), & W > E_{SMX} \\[4pt] \lambda_{ES}\,(E_{SMN} - E_S), & W < E_{SMN} \end{cases} && \text{(marrow RBC‑production rate \(E_S\))} \end{align} \]

Malaria Example 2

\[ \begin{align} \dot{\mathrm{Act}} &= H(S_{\mathrm{Act}}) - \lambda_{\mathrm{Act}}\,\mathrm{Act} && \text{(innate actuator)} \\[4pt] \dot{\mathrm{Att}} &= H(S_{\mathrm{Att}}) - \lambda_{\mathrm{Att}}\,\mathrm{Att} && \text{(innate attacker)} \\ \dot{\mathrm{Act}} &= H(S_{\mathrm{Act}}) - \lambda_{\mathrm{Act}}\,\mathrm{Act} && \text{(adaptive actuator)} \\[4pt] \dot{G}_1 &= H\!\bigl(\lambda_{\mathrm{Act}}\mathrm{Act}-S_{G,\mathrm{th}}\bigr) - k_G G_1 && \text{(growth‑phase compartment \(G_1\))} \\[4pt] \dot{G}_n &= k_G G_{n-1} - k_G G_n && \text{(growth‑phase compartment \(G_n\), $2\le n\le N_{cG}+1$)} \\[4pt] \dot{\mathrm{Att}} &= H(S_{G,\mathrm{Att}}) - \lambda_{\mathrm{Att}}\,\mathrm{Att} && \text{(adaptive attacker)} \end{align} \]

Malaria Example 2

Peak (\(I_{PK}\)) and integrated (\(I_{INT}\)) parasitemia as a function of model parameters.

Worm Infection Example 1

“Modelling within‑host parasite dynamics of schistosomiasis” by Chiyaka et al 2010 CMMM.

Question: How can within-host parasite dynamics be controlled by different parts of the immune response?

Approach: Medium-size mathematical model explored for different parameter values.

Worm Infection Example 1

\[ \begin{align} \dot{L} &= \lambda\,f(L,E) - \bigl(m_L + \mu_L + d_L\bigr)\,L, && \text{(larval stage \(L\))}\\[4pt] \dot{W}_I &= m_L\,L - \bigl(m_I + \mu_I + d_I\bigr)\,W_I, && \text{(immature worms \(W_I\))}\\[4pt] \dot{W}_P &= \tfrac{1}{2}m_I\,W_I - \bigl(m_P + \mu_P\bigr)\,W_P, && \text{(paired adult worms \(W_P\))}\\[4pt] \dot{E} &= m_P\,N_E\,W_P - \bigl(m_E + \mu_E\bigr)\,E, && \text{(eggs \(E\))}\\[8pt] \dot{M}_R &= s_R + r_R\bigl(L + W_T + E\bigr)M_R && \text{(resting macrophages \(M_R\))}\\[4pt] & - s_A\bigl(L + W_T + E\bigr)M_R + \beta_A\,M_A - \mu_R\,M_R, \\ \dot{M}_A &= s_A\bigl(L + W_T + E\bigr)M_R - \beta_A\,M_A - \mu_A\,M_A, && \text{(activated macrophages \(M_A\))}\\[4pt] \dot{T} &= s_T + r_T\bigl(E + M_R + M_A\bigr)T - \mu_T\,T. && \text{(T cells \(T\))} \end{align} \]

More equations for model variants discussed in the paper.

Worm Infection Example 1

Time series of model dynamics.

Worm Infection Example 2

“Within-host mechanisms of immune regulation explain the contrasting dynamics of two helminth species in both single and dual infections”, Vanalli et al, 2020 PLoS Comp Bio.

Question: How does the interaction between parasites and the immune response lead to different infection outcomes for single or coinfections of rabbits with Trichostrongylus retortaeformis and Graphidium strigosum?

Approach: 16 variants of fairly simple mathematical models were explored and fit to data to distinguish different immune processes/hypotheses.

Worm Infection Example 2

\[ \begin{align} \dot P_i &= \sigma_i L_{0i} e^{-k_i t} \;-\; \mu\,P_i \;-\; \alpha_i I_{1i} P_i, && \text{parasite intensity}\\[4pt] \dot I_{1i} &= \beta_{1i}\, I_{1i}^{a_i} I_{2i}^{c_i} P_i^{d_i} \;-\; \delta_{1i} I_{1i} \;+\; \Lambda_{1i}, && \text{species-specific IgA}\\[4pt] \dot I_{2i} &= \beta_{2i}\, I_{2i}^{b_i} P_i \;-\; \delta_{2i} I_{2i} \;+\; \Lambda_{2i}, && \text{IL4 response} \end{align} \]

i indexes the pathogen type.

Worm Infection Example 2

\[ \begin{align} \dot P_{\mathrm{TR}} &= \sigma_{\mathrm{TR}} L_{0,\mathrm{TR}} e^{-k_{\mathrm{TR}} t} - \mu P_{\mathrm{TR}} - \alpha_{\mathrm{TR}} I_{1,\mathrm{TR}} P_{\mathrm{TR}} - \alpha_{\mathrm{GS},\mathrm{TR}} I_{1,\mathrm{GS}} P_{\mathrm{TR}} \\[4pt] \dot P_{\mathrm{GS}} &= \sigma_{\mathrm{GS}} L_{0,\mathrm{GS}} e^{-k_{\mathrm{GS}} t} - \mu P_{\mathrm{GS}} - \alpha_{\mathrm{GS}} I_{1,\mathrm{GS}} P_{\mathrm{GS}} - \alpha_{\mathrm{TR},\mathrm{GS}} I_{1,\mathrm{TR}} P_{\mathrm{GS}} \\[4pt] \dot I_{1,\mathrm{TR}} &= \beta_{1,\mathrm{TR}} I_{1,\mathrm{TR}}^{a_{\mathrm{TR}}} I_{2,\mathrm{TR}}^{c_{\mathrm{TR}}} P_{\mathrm{TR}}^{d_{\mathrm{TR}}} + \beta_{1,\mathrm{GS},\mathrm{TR}} I_{2,\mathrm{GS}}^{c_{\mathrm{GS},\mathrm{TR}}} - \delta_{1,\mathrm{TR}} I_{1,\mathrm{TR}} + \Lambda_{1,\mathrm{TR}} \\[4pt] \dot I_{1,\mathrm{GS}} &= \beta_{1,\mathrm{GS}} I_{1,\mathrm{GS}}^{a_{\mathrm{GS}}} I_{2,\mathrm{GS}}^{c_{\mathrm{GS}}} P_{\mathrm{GS}}^{d_{\mathrm{GS}}} + \beta_{1,\mathrm{TR},\mathrm{GS}} I_{2,\mathrm{TR}}^{c_{\mathrm{TR},mathrm{GS}}} - \delta_{1,\mathrm{GS}} I_{1,\mathrm{GS}} + \Lambda_{1,\mathrm{GS}} \\[4pt] \dot I_{2,\mathrm{TR}} &= \beta_{2,\mathrm{TR}} I_{2,\mathrm{TR}}^{b_{\mathrm{TR}}} P_{\mathrm{TR}} - \delta_{2,\mathrm{TR}} I_{2,\mathrm{TR}} + \Lambda_{2,\mathrm{TR}} \\[4pt] \dot I_{2,\mathrm{GS}} &= \beta_{2,\mathrm{GS}} I_{2,\mathrm{GS}}^{b_{\mathrm{GS}}} P_{\mathrm{GS}} - \delta_{2,\mathrm{GS}} I_{2,\mathrm{GS}} + \Lambda_{2,\mathrm{GS}} \end{align} \]

Worm Infection Example 2

Model parameter table.

Worm Infection Example 2

Model fits to data.

Worm Infection Example 2

Model comparison table.

Summary

  • There is nothing fundamentally different between macroparasite and virus/bacteria models.
  • What is included and excluded in the model, and which processes are modeled, depends on the system, question and choices made by the modeler.