Mathematical Modelling of Infection Dynamics of Animal African Trypanosomiasis

Introduction

African trypanosomes, also known as old world trypanosomes, are single-celled protozoan hemoflagellates, especially of the genus Trypanosoma, that live in the mammalian bloodstream, causing widespread disease in humans and livestock. Nagana or Animal African Trypanosomiasis, which affects wild and domestic animals such as cattle, sheep, goats, and horses in 40 countries in sub-Saharan Africa, is caused by trypanosome species such as Trypanosoma brucei, which has been further grouped into three subspecies: Trypanosoma brucei brucei, Trypanosoma brucei gambiense, and Trypanosoma brucei rhodesiense. Trypanosoma brucei brucei doesn’t infect humans under normal conditions but infects cattle and other animals, while the latter two cause disease in humans. Trypanosomiasis has a significant impact on health and economic conditions within affected areas. It causes major economic loss to the development of agriculture as it reduces meat and milk production, as well as a reduction in draught animal production.

These protozoa are transmitted to the host via the bite of tsetse fly. Once the protozoa enter the bloodstream of the animals, they cause fever, weakness, lethargy, and anemia, leading to weight loss and a reduction in milk production.

They also reduce the fertility of the affected livestock and may also lead to their death. Trypanosomes can survive for a long period of time in the host and establish chronic infection by using immune evasion strategies, which enhance their dissemination and transmission.

These parasites proliferate in the bloodstream and tissue spaces of the host where they are exposed to the host's immune surveillance. They survive in this hostile environment by using their immune evasion strategy, which is antigenic variation.

Trypanosomes are covered by a thick protein coat made of variant surface glycoprotein (VSG) which protects them from the host's immune response. The trypanosomes possess about 2000 silent VSG genes which code for different forms of surface proteins or antigen molecules. These variants differ from each other, usually at either one or more epitopes, which are recognition sites of antigens for stimulation of specific immune response. Variant-specific antibodies are produced by the host's immune system after recognizing the glycoprotein coat. These antibodies destroy the parasites present in the host, but most of the parasites evade the antibody responses by changing their surface coat, resulting in the formation of new VSGs that are no longer recognized by the previously produced antibodies. The parasites with new VSGs proliferate until the production of new antibodies that are capable of fighting with the parasites with the new VSGs coat, resulting in multiple waves of parasitemia.

Trypanosomes differentiate between different life stages as they encounter different environments while transmitting from the blood of a mammalian host to the midgut of tsetse fly and then migrating to the salivary gland before entering a new host via the vector's bite in their life cycle. These different life stages help the parasite in survival as they are challenged by the various environments. Its life cycle is differentiated into two stages: bloodstream stage and tsetse stage. During the bloodstream stage, there is a transition from the proliferative slender cells to non-proliferative stumpy cells with transitional intermediate cells. Slender cells have replication capacity as well as antigenic variation capacity, which enables them to maintain parasitemia as well as evade the immune response. The proliferative slender cells differentiate to non-proliferative stumpy forms through a Quorum sensing-like mechanism, whereby a soluble parasite-derived factor triggers the production of stumpy form. This factor is called stumpy induction factor (i.e., SIF), produced by slender cells, not by cells in the stumpy class. As the density of slender cells increases, SIF accumulates, triggering the differentiation of uncommitted slender cells to committed slender cells, which show commitment to cell division before differentiating to irreversibly cell cycle-arrested stumpy cells. The stumpy cells so formed are important for transmission as they can endure the stress of tsetse uptake. To maximize its survival and dissemination, they keep a balance between transmission, pathogenicity, and its proliferation by regulating their growth within-host in a density-dependent manner.

Once their density passes a critical threshold, slender cells differentiate to stumpy cells that can only infect tsetse fly, leading to the restriction of the growth of the trypanosomes' population and predominance of transmission stages. To ensure their transmission, stumpy cells have a mechanism known as endocytosis in which there is a movement of VSG-bound antibodies to the flagellar pocket (site of endocytosis), leading to antibody clearance and thus inhibiting complement lysis of the parasite. Stumpy cells have a greater rate of endocytosis than slender cells, resulting in stumpy cells remaining in the host for a longer duration, exhibiting predominance of transmission stages in the late phase of infection. Thus, the infection dynamics of the parasites are controlled by a balance between slender and stumpy form, and density-dependent transition from slender to stumpy form is controlled by SIF, a parasite-derived factor that acts via the CaMP pathway. We don't have much information regarding the molecular mechanism of SIF receptors. In this, I will write the objective.

Previous Models and Their Shortcomings

To understand the within-host infection dynamics of Animal African Trypanosomiasis, people came up with different mathematical models. Most of the models used density-dependent differentiation to describe slender-to-stumpy transition rather than SIF concentration-dependent differentiation. The first model to be based on SIF-induced differentiation was proposed by Savil and Seed. Their model was given by:

dL1(t) = a1L1(t) * w(t)L1(t) dt

dL2(t) = a1L2(t) + w(t)L1(t) * exp(a1t2) * w(t-t2)L1(t-t2) dt

dL3(t) = exp(a1t2) * w(t-t2)L1(t-t2) * exp(a1t2) * w(t-t2-t3)L1(t-t2-t3) dt

dS(t) = exp(a1t2) * w(t-t2-t3)L1(t-t2-t3) - a4S(t) dt

dF(t) = L1(t) + L2(t) - a5F(t) dt

where w(t) = a2 + a3F(t)

Where L1, L2, L3, and S represent the concentration of dividing slender cells not committed to differentiate, dividing slender cells committed to differentiate, non-dividing slender cells committed to differentiate, and stumpy cells respectively, and F denotes the concentration of SIF. They considered two differentiation mechanisms: SIF-induced differentiation and background differentiation at rates a3 and a2. The SIF degradation rate and removal rate of stumpy cells by natural cell death are represented by a5 and a4, and t1, t2, t3 are the time spent by the parasite from one stage to another. Their model didn't incorporate the effects of the host immune system. Their model gave a good fit to the data collected from previous experiments in immunosuppressed mice. The later study done by Mathew et al. incorporated the effect of the host immune response in SIF-induced differentiation mathematical model. The switching of the parasite was considered to be sequential, and a simple model of the immune response was used due to a lack of data related to the immune response.

One model proposed by Erida considered density-dependent differentiation with stochastic switching. The dynamics of slender cells (vi), stumpy cells (mi), and specific immune response (ai) are given by the following equations:

dvi(t) = rvi(1 - (V + M)/K) * daivi dt

dmi(t) = rvi((V + M)/K) * i * mi dt

dai(t) = c * x * davv(t) + m(t) * i * (1 - ai) * dt

where growth of slender cells is denoted by rv, and the natural cell death rate of stumpy cells is denoted by m. The cells in each class are cleared by antibodies at rates d and . The differentiation rate is rvi(1 - (V + M)/K) where K represents the total carrying capacity, V = vi, and M = mi. The growth rate of antibodies depends on different parameters, including delay time, sensitivity x, rate of growth of specific immune response c, and threshold level C above which there is maximal growth of the immune response. The switching was assumed to be stochastic because the new variant emerges in a small number and it may be prone to extinction.

By combining the ideas from previous models, we came up with a new model of within-host infection dynamics of Animal African Trypanosomiasis. We have proposed a mathematical model to understand the mechanism that maintains the chronicity of infection and those driving antigenic variation over the period of trypanosome infection.

Our model included the effect of the host immune response, which was not considered in the Seeds paper. Different from Mathew's paper is that they assumed activation of the initial immune response after a certain time (delay time) only for the first variant but not for the other emerging variants. Our approach is similar to Erida; we have tried to incorporate a delay between immune responses for all the variants to represent the time taken by the immune system to respond to various emerging variants. They also assumed the switching to be sequential, in which the first variant switches to the second and the second variant switches to the third and so on with different switching rates, while we have assumed switching to be non-sequential, which may be more biologically correct. We have considered the switching to be stochastic, which doesn't have any contribution towards parasite growth but is a way of generating antigenic variants that were absent previously. During the course of infection, a new variant emerges in a small number and it has more chances of going extinct, so switching is assumed to be stochastic. If the mean activation rate is less than the critical activation rate, then that variant will never be generated during the period of infection (Gini), whereas this aspect is absent in Mathew's paper where the infection dynamics are continuous.

Hesse et al (1995) suggested that the mediator of the quorum sensing mechanism is a trypanosome-derived factor. So, we incorporated SIF-induced differentiation rather than density-dependent differentiation in our model. Our main departure from the approach of Gini et al (2010) is assuming the rate of differentiation to be proportional to soluble parasite-derived factor (SIF-induced differentiation) rather than the density of the parasite. We will further investigate how the dynamics of SIF-dependent differentiation model is different from the density-dependent differentiation model.

New Model

We propose a mathematical model to study the within-host infection dynamics of Animal African Trypanosomiasis. Our model is an extension of the model proposed by Erida et al. (2010), that includes parasite-derived soluble factor (SIF)-dependent differentiation rather than density-dependent differentiation. We have considered three distinct cell types in our model: uncommitted slender cells, committed slender cells, and stumpy cells. The concentration of SIF is represented by 'f'. The number of uncommitted slender cells, committed slender cells, and stumpy cells are denoted by 'vi', 'li', and 'mi', respectively, and 'ai' being the acquired immune response for any variant 'i', where 'i' varies from 1 to 'N'. 'N' represents the total number of variants present in the genetic archive.

The dynamics of uncommitted slender cells and immune response is given by the following equation:

rate of change of 'vi' = cell division - SIF-induced differentiation - immune clearance

dvi/dt = rvvi - b1fvi - b2vi dvaivi

The density of uncommitted slender cells depends on its replication rate, rate of differentiation, and immune clearance rate. The uncommitted slender cells replicate at a rate 'rv', which is constant throughout the infection. Two important differentiation mechanisms considered in our model are SIF concentration-dependent differentiation and background differentiation. We assume SIF concentration-dependent differentiation to be linear, which implies a differentiation rate being proportional to the concentration of SIF because a non-linear SIF concentration-dependent differentiation rate gives a poor fit to the data (JR Seed). Soluble factor-dependent differentiation is independent of background differentiation as stumpy cells are present even during the growth phase of the infection when the SIF concentration has not reached a point to induce differentiation of uncommitted slender cells, implying background differentiation happens even before SIF concentration-dependent differentiation. SIF-induced differentiation rate and background differentiation rate are denoted by 'b1' and 'b2', which remain constant throughout the infection. The cells in the uncommitted class move out by forward differentiation and clearance through host immune response. Upon receiving the differentiation signal, uncommitted slender cells differentiate to committed slender cells, which is an instantaneous transition. The rate at which uncommitted slender cells get cleared by antibody response 'ai' is denoted by 'dv'.

The dynamics of committed slender cells and immune response is given by:

rate of change of 'li' = cell division - differentiate committed cell - immune clearance - transformation to stumpy form

dli/dt = rlli + b1fvi + b2vi dlaili + li

Birth rate of committed slender cells and the rate at which these cells differentiate to stumpy cells is denoted by 'rl' and ',', respectively, which remains constant throughout the infection. These cells are cleared from the host via antibodies at a rate 'dl'. We have assumed that committed slender cells undergo three rounds of cell division before undergoing cell cycle arrest and morphological transformation to stumpy cells.

Dynamics of the rate of change of SIF concentration:

rate of change of 'SIF' = 'SIF production' - 'SIF degradation'

df/dt = V + M - b3f

where V = vi and M = mi. The slender cells grow exponentially and release SIF. The rate at which SIF is produced is proportional to the population of slender cells. SIF is removed from the system exponentially at rate 'b3'. We don't have much information related to SIF's production rate and about the level of concentration at which it induces differentiation. To overcome this, we did normalization of the SIF production rate (see Appendix A).

Dynamics of stumpy cells and immune response is given by:

rate of change of 'mi' = transformed stumpy cells - immune clearance - natural death

dmi/dt = li - mi dmmi

The committed slender cells differentiate to stumpy cells at rate ','. The loss of stumpy cells is due to natural cell death and immune response at rates 'and 'dm', respectively.

Kinetics of acquired immune response is given by:

da/dt = vi(t) - mi(t) xi = c - i(1 - ai)/C

Where 't' is the time taken by the variants for specific antibody response stimulation, 'x' is the sensitivity of the specific antibody response to small antigen stimulation, 'c' is the rate of growth of antibody response, and 'C' is the threshold of the variant population level which leads to the maximum growth of specific antibody response.

The system of equations representing the within-host infection dynamics for any variant 'i' are as follows:

dvi/dt = rvvi - b1fvi - b2vi dvaivi (1)

dli/dt = rlli + b1fvi + b2vi dlaili + li (2)

df/dt = V + M - b3f (3)

dmi/dt = li - mi dmmi (4)

da/dt = vi(t) - mi(t) xi = c - i(1 - ai)/C (5)

We have followed the same approach as that of Erida for the emergence of a new variant of the parasite, in which switching to a new variant is assumed to be stochastic as the new antigenic variants emerge in a very small number and are likely to go extinct. We assume 'Pi(t)' to be the probability that the ith antigenic variant has not been generated by time 't'. Differently from Erida, we assume the rate of change of 'Pi(t)' depends only on uncommitted slender cell's population as only cells in the uncommitted class can switch to new variants. The 'Pi(t)' dynamics is given by:

dPi/dt = Pi Σj≠i sji vj (6)

Where 'sji' is the rate at which variant 'i' switches to variant 'j'.

First arrival time for an arbitrary variant 'i' (ti) is calculated by choosing a random number 'r' from a uniform distribution (0, 1), ith variant jumps from 0 to 1 at the time when 'Pi(t)' reaches 'r'. After this point, dynamics of antigenic variant and acquired immune response is given by equation 1 to 5. The dynamics of uncommitted slender variant is as follows:

dvi/dt = { rvvi - b1fvi - b2vi dvaivi + 1; t < ti, 0; t ≥ ti }

All model parameters are listed in table 1.

Results

We conducted a stability analysis of the model. The results of our analysis show that if ai > (rv * b2) / dv, then the infection clearance state is stable, and the disease dies out. There is also an infection persistence state (v, l, m, f, a) that exists when dl * rl > 0 and (rv * b2 * dv * b3) > 0, and it is stable when (dl * rl) + b3 > (rv * b2 * dv * b3), and the disease persists.

Table 1: Model Parameters and Interpretations

Parameters	Definition	Dimensional Value	Reference
rv	Birth rate of the noncommitted slender cells (h^-1)	0.33	Savill and Seed 2003
rl	Birth rate of the committed slender cells (h^-1)	0.33	Savill and Seed 2003
b1	SIF-induced differentiation rate (slender cells)(h^-1)	0.5 - 3 x 10^9	Savill and Seed 2003, Matthews
b2	Background differentiation rate of slender cells (h^-1)	0.15	Savill and Seed 2003
b3	SIF degradation rate (h^-1)	0.2-1.4	Savill and Seed 2003, Matthews
dv	Maximum killing efficiency of noncommitted slender cells by the immune response	0.5	McLintock et al. 1993
dl	Maximum killing efficiency of committed slender cells	0.5	McLintock et al. 1993
dm	Maximum killing efficiency of stumpy cells	0.1	McLintock et al. 1993
c	Rate of growth of specific immune response	100	Lythgoe et al. 2007
C	Threshold variant population level leading to maximal growth of specific immune response	10^8 - 10^12	Lythgoe et al. 2007, varied
x	Sensitivity of immune responses to small parasite concentration	1-3	Lythgoe et al. 2007, varied
	Differentiation of slender to stumpy form (stumpy cell mortality rate)	2.5 x 10^-2	Tyler et al. 2001; Savill and Seed 2003
	Delay in the stimulation of specific immunity (hours)	100	Tyler et al. 2001; varied

Discussion

In this study, we have developed a mathematical model to investigate the within-host infection dynamics of Animal African Trypanosomiasis, a devastating disease caused by protozoan hemoflagellates of the genus Trypanosoma. Our model builds upon previous models that attempted to understand the dynamics of trypanosome infection within the host, particularly focusing on the mechanisms driving antigenic variation and immune response.

One of the key contributions of our model is the incorporation of the host immune response, which was not adequately considered in some previous models. We have introduced a more detailed representation of the immune response by including the dynamics of acquired immune response for different antigenic variants. This addition enhances our understanding of how the host's immune system interacts with the trypanosome population.

Furthermore, we have addressed the mechanism of parasite differentiation within the host. Unlike some earlier models that primarily relied on density-dependent differentiation, we have incorporated soluble parasite-derived factor (SIF)-dependent differentiation, which is believed to play a crucial role in trypanosome differentiation. This modification provides a more biologically relevant framework for understanding the transition of slender cells to stumpy cells, which are important for transmission.

Our model also accounts for the stochastic nature of antigenic variation, where new variants emerge in small numbers and may go extinct. This stochastic switching mechanism is more realistic and captures the inherent variability in the generation of antigenic variants.

Conclusion

In conclusion, our mathematical model represents a significant advancement in understanding the within-host infection dynamics of Animal African Trypanosomiasis. By incorporating the host immune response, SIF-dependent differentiation, and stochastic switching, we have developed a more comprehensive and biologically relevant framework for studying this complex disease.

Our model provides insights into the factors that contribute to the chronicity of infection and the mechanisms behind antigenic variation. It offers a valuable tool for researchers and policymakers working on strategies to control and combat this devastating disease, which has significant implications for both human and livestock health in sub-Saharan Africa.

Future research can build upon our model to explore various scenarios and interventions for controlling trypanosome infections, ultimately contributing to the development of more effective strategies for managing this important public health and economic challenge.