Antibody Dynamics for Plasmodium vivax Malaria: A Mathematical Model

Abstract

Malaria is a mosquito-borne disease that, despite intensive control and mitigation initiatives, continues to pose an enormous public health burden. Plasmodium vivax is one of the principal causes of malaria in humans. Antibodies, which play a fundamental role in the host response to P. vivax, are acquired through exposure to the parasite. Here, we introduce a stochastic, within-host model of antibody responses to P. vivax for an individual in a general transmission setting. We begin by developing an epidemiological framework accounting for P. vivax infections resulting from new mosquito bites (primary infections), as well as the activation of dormant-liver stages known as hypnozoites (relapses). By constructing an infinite server queue, we obtain analytic results for the distribution of relapses in a general transmission setting. We then consider a simple model of antibody kinetics, whereby antibodies are boosted with each infection, but are subject to decay over time. By embedding this model for antibody kinetics in the epidemiological framework using a generalised shot noise process, we derive analytic expressions governing the distribution of antibody levels for a single individual in a general transmission setting. Our work provides a means to explore exposure-dependent antibody dynamics for P. vivax, with the potential to address key questions in the context of serological surveillance and acquired immunity.

Appendix

Recall that $\varvec{\omega }(t)$ denotes the antibody response to a panel of proteins $V_1, \dots , V_R$ time t after a single infection, with

$$\begin{aligned} \varvec{\omega }(t) = (\eta _{1} e^{-b_1 t}, \dots , \eta _R e^{-b_R t}) \end{aligned}$$

from Eq. (9), with $b_i, \eta _i > 0$ for each i.

Given that the distribution of $\eta _i$ decays exponentially for each i, the MGF of $\varvec{\omega }(t)$ is well defined in an open hypersphere around the origin. We seek to show that the inequality

$$\begin{aligned} \frac{\nu }{\nu +1}\Bigg [1 - B(t) + \int ^t_0 \Big ( G(u) {\mathbb {E}} \big [e^{ \varvec{\omega }(t - u) \cdot \varvec{s}} \big ] \Big ) \mathrm{d}u \Bigg ] < 1 \end{aligned}$$

(39)

also holds for an open hypersphere around $\varvec{s}=\varvec{0}$.

By noting that $G(u)=\mathrm{d}B(u)/\mathrm{d}u$ (Eq. 13), condition (39) can equivalently be written

$$\begin{aligned} \int ^{t}_0 G(u) \Big ( {\mathbb {E}} \big [e^{ \varvec{\omega }(t - u) \cdot \varvec{s}} \big ] - 1 \Big ) \mathrm{d}u < \frac{1}{\nu }. \end{aligned}$$

(40)

Since $\varvec{\omega }(t) \le \varvec{\omega }(0)$ for all $t \ge 0$,

$$\begin{aligned} \int ^{t}_0 G(u) \Big ( {\mathbb {E}} \big [e^{ \varvec{\omega }(t - u) \cdot \varvec{s}} \big ] - 1 \Big ) \mathrm{d}u \le B(t) \Big ( {\mathbb {E}} \big [e^{ \varvec{\omega }(0) \cdot \varvec{s}} \big ] - 1 \Big ) \le {\mathbb {E}} \big [e^{ \varvec{\omega }(0) \cdot \varvec{s}} \big ] - 1, \end{aligned}$$

noting that the probability mass $B(t) \in [0, 1]$ for all $t \ge 0$.

By assumption, ${\mathbb {E}} \big [e^{ \varvec{\omega }(0) \cdot \varvec{s}} \big ]$, the MGF of $\varvec{\omega }(0) = (\eta _1, \dots , \eta _R)$ is well defined in some open hypersphere around the origin. It follows that ${\mathbb {E}} \big [e^{ \varvec{\omega }(0) \cdot \varvec{s}} \big ]$ is differentiable, therefore continuous at the origin with respect to $\varvec{s}$, hence, there exists $\delta >0$ such that

$$\begin{aligned} \Big |{\mathbb {E}} \big [e^{ \varvec{\omega }(0) \cdot \varvec{s}} \big ] - 1 \Big |\le \frac{1}{\nu } \text { for all } |\varvec{s} |< \delta , \end{aligned}$$

that is, the condition specified in Eq. (39) is satisfied in an open hypersphere around the origin, as required.