# Epidemiological Modelof Chagas Disease

## Introduction

We generated a model of the disease progression of Chagas disease in a human population. After equilibrium had been reached, we introduced our diagnostic device for congenital Chagas to assess the effectiveness of diagnosing congenital Chagas disease. We hoped to investigate the effect that curing infected newborns would have on the numbers of infected individuals in the whole population. The model parameters reflect the country of Bolivia, where congenital Chagas disease is a major issue.

We approached Professor Mike Bonsall to gain a better understanding of the principles of disease modelling, and to equip us with the skills to create our own epidemiological model for Chagas disease.

## Methods

### Technique

Our model combines aspects of SIR (Susceptible, Infected, Recovered) and vector disease modeling to appropriately represent Chagas disease, which can be transmitted via a vector (Triatomine), horizontally (e.g. via blood transfusions and bodily fluids), and vertically (infected mother to child).

The parameters of our model reflect the country of Bolivia. The implementation of the diagnostic device, at t = 0 years, is modelled by a change in rIab, reflecting that 70% of infected newborns become diagnosed and treated, as compared to 0% prior to our diagnostic device.

### Assumptions

1. All infants are tested with our diagnostic device
2. 70% of diagnosed infected infants successfully complete treatment (ref. E)
3. Birth and mortality rates in human and vector populations are fixed
4. Infected women in the acute phase cannot give birth
5. There is only one species of Triatomine (Rhodnius prolixus)
6. The impact of non-human host species (e.g. dogs, cats, and other synanthropic animals) is negligible
7. Frequency dependent transmission: 1 vector can infect only 1 host at a time
8. There is a fitness difference between healthy and infected vectors
9. The carrying capacity of the vector is 10x the carrying capacity of humans
10. Infants are cured instantaneously upon receiving treatment
11. No vector or transfusion control measures are undertaken in the duration of our model
12. No infants are diagnosed and treated prior to the implementation of our diagnostic device
13. The rate of movement from acute to chronic phase is not age-dependent

### Parameters

Table 1. Parameters for Chagas Disease Epidemiological Model

Parameter Variable Name Value Reference*
Carrying capacity of Triatomine $$K_{b}$$ $$K_{p}~{\times}~10~(1{\times}10^{9})~Triatomine$$ A
Birth rate of Triatomine $$r_{b}$$ $$36~births~Triatomine^{-1}~year^{-1}$$ A
Fitness of infected vector to give birth1 $$f_v$$ $$0.75$$ A
Mortality rate of vector $$d_b$$ $$1.73~deaths~Triatomine^{-1}~year^{-1}$$ A
Probability of infection from human in acute phase $$c_a$$ $$0.61$$ A
Probability of infection from human in chronic phase $$c_d$$ $$0.2$$ A
Carrying capacity of humans2 $$K_p$$ $$1\times10^{8}~humans$$
Birth rate of humans $$r_{p}$$ $$0.023549~births~human^{-1}~year^{-1}$$ B
Mortality rate of humans $$d_{n}$$ $$0.007353~deaths~human^{-1}~year^{-1}$$ C
Additional mortality rate of chronically infected humans $$d_{d}$$ $$0.172647~deaths~human^{-1}~year^{-1}$$ D
Contact rate of vectors and humans $$\beta$$ $$41$$ A
Probability of infection from vector to susceptible human $$c_{vn}$$ $$0.00058$$ A
Probability of infection from infected chronic mother to child $$c_{v_{c}}$$ $$0.049$$ E
Proportion of acute phase patients cured $$r_{Ia}$$ $$0.65$$ F
Treatment completion rate for newborns $$r_{I_{ab}}$$ $$0.7$$ E
Movement from acute phase to chronic phase $$\delta$$ $$0.3$$ -
Immunity levels of treated adults3 $$\theta$$ $$\gt 0.03$$
Immunity levels of treated newborns5 $$\theta_{2}$$ $$\gt 0.03$$

*See reference code at the bottom of the document.

10.75 is an estimate based on the suggested value of 0-1 by ref. A.

2An estimate; 10 times the current population of Bolivia.

3The immunity level is not currently known; Dr. Yves Carlier, an expert in Chagas disease, believes there is a high probability of immunity after the treatment of acutely infected Chagas patients. A parameter scan of our model indicates that our diagnostic device will be effective at lowering Chagas disease prevalence if the immunity value is greater than 0.03.

### Equations

$$B$$ = uninfected vectors

$$V$$ = infected vectors

$$S$$ = susceptible adults

$$I_{a}$$ = infected adults in the chronic phase

$$I_{b}$$ = infected babies in the acute phase

$$I_{c}$$ = infected individuals in the chronic phase

$$R$$ = immune adults

$$C$$ = cured babies

$$D$$ = death due to Chagas disease

$$Checked$$ = tested babies

$$Rb$$ = immune babies

$$Sb$$ = susceptible babies

$$N = S + Ia + Ib + Ic + R$$

### Sensitivity Analysis

We conducted a sensitivity analysis to analyze the impacts of uncertainties in parameter values on our model outcomes. The figure below shows the normalized sensitivities – the ratio of the relative change of the output (x-axis) to the relative change of the parameter (y-axis). Please refer to the parameters and equations above (Table 1., in the 'Technique, Assumptions, Parameters, Equations' collapsible box) for an explanation of the parameter/output names; those not included above are '%chg I' (percent change in the total number of infected people), '1-person eq' (the equilibrium point, defined as the time-point when the change in the number of infected people is less than one person per year), and '5% eq' (the time-point at which 95% of the total change in infected people, as calculated up to the equilibrium point, has been reached). We have included outputs for the data sets to which our diagnostic intervention has (output_{yes}) and has not (output_{no}) been applied.

Larger values (more yellow) correspond to greater sensitivity of an output on a parameter.

###### Figure 2. A Sensitivity Analysis of Parameters in the Epidemiological Model

This analysis indicates that the results of the model (in particular the percent change in the total number of infected people, %chg I) are most sensitive to changes in the birth rate of humans ($$r_{p}$$), death rate of humans ($$d_{n}$$), and immunity rate ($$\theta$$). As explained in more detail in the evaluation section, below, we are confident in the former two parameter values. There is great uncertainty surrounding the value of the latter parameter, so we conducted a parameter scan of immunity rate.

### Parameter Scan

The parameter scan quantifies the impact of the value of immunity rate on our model outcomes. We ran the model with immunity rates ranging from 0 to 1 and plotted these values against the resulting percent change in the total number of infected people. As long as the immunity rate is higher than 0.0312, there will be a decrease in the number of infected people and our diagnostic will be beneficial. Please see our evaluation below for greater analysis of the parameter scan results.

## Results

The implementation of our diagnostic device has a profound effect on the spread of Chagas disease in the population modelled after Bolivia. 100 years after our diagnostic intervention, 125,000 people would already have been spared from getting Chagas disease. By 150 years, this number will have reached 134,000. This high value is extremely important, as it represents an amelioration in the lives of Latin American people. The effect of our diagnostic device will also have strong economic benefits: currently, a lot of money is being lost to the direct health-care costs and indirect costs of decreased worker productivity associated with Chagas disease. The indirect costs can be represented in disability-adjusted life years (DALYs), which are a representation of years lost due to living with a disability (YLD) and due to premature mortality (YLL). Our diagnostic device will decrease the number of Chagas cases and therefore DALYs, improving the lives of thousands of people and saving the economy significant amounts of money.

## Discussions

Assuming the implementation of our diagnostic device results in the diagnosis and subsequent treatment of 70% of infected newborns, there is a noticeable decrease in the number of acutely, chronically, and total infected humans in our model. This is likely a result of the decreased number of infectious newborns in the population. The treatment of newborns prevents their ability to infect others (horizontally or vertically) over the course of their life and may decrease the susceptible pool size – once treated, the individual is cured and immune for life. By implementing our inexpensive and easy-to-use diagnostic device on a large scale in the modeled population, we would be able to drastically decrease congenital cases of Chagas disease and thus the spread of the Chagas disease to the rest of the population.

## Evaluation

The predictive value of our model is dependent on the accuracy of the many parameters considered. All parameter values were informed by research papers and credible websites published/updated within the last 10 years. There is, however, some uncertainty surrounding the parameters of vector fitness and immunity. A sensitivity analysis shows that the value of the fitness of an infected vector to give birth does not greatly affect the outcome of the model. Immunity rate has a larger impact, but a parameter scan indicated that our diagnostic device would be successful in lowering Chagas disease cases for an immunity rate value greater than 0.03 (corresponding to a 3% lower chance in getting infected again after successful treatment than a susceptible person who has never been infected). We consulted Dr. Yves Carlier, an expert in Chagas disease; he believed it was likely that treatment of Chagas disease in newborns would confer some immunity, strengthening our confidence in our device and model. The sensitivity analysis also demonstrated that the parameters whose values were most likely impact to the outcome of our model are the birth and death rates of humans. We are very confident in these parameters, which are taken from the World Bank database.

The assumptions we made in our model also contribute to its predictive value. We made the assumptions, stated above, in cases where there was insufficient data to include a parameter and, upon the recommendation of our supervisors, to limit the complexity of the model. Many of these are backed up by extensive research and literature (as referenced in Table 1., in the 'Technique, Assumptions, Parameters, Equations' collapsible box). There are, however, a few assumptions that may limit the predictive accuracy of our model. These relate to biology, sociology, and treatment:

• Biology: Many animals that can transmit Chagas disease, including synanthropic animals and additional species of Triatomine, are not considered in this model. They were excluded for reasons of clarity and their comparatively small contribution to Chagas disease prevalence. The inclusion of these would likely increase the number of Chagas cases in our model.
• Sociology: Our model assumes that all infants are tested (but only 70% treated). While there are incentives to give birth in hospitals – as Dr. Cristina Alonso-Vega, the leader of the Program for National Control of Congenital Chagas in Bolivia (2004–2009) informed us – some women still deliver their babies at home. These newborns would have a lower likelihood of being tested for Chagas disease. Another factor that should be, but is not explicitly considered in our model, is the dependence of parameters on how rural/urban an area is.
• Treatment: We assumed that infants are cured instantaneously upon receiving treatment. This is not physiologically true – it takes 6 weeks in real life. We made the assumption, however, because 6 weeks is a negligible time with respect to the duration of our entire model (200 years).

Many of our assumptions are realistic at the time the model was created, but may change in the future. One example is the treatment completion rate of newborns (70%) (ref. E); we hope this number will increase in the future.

In the future, we could expand on our model by including more parameters and adjusting some of the assumptions discussed above. We would also continue to update parameter values to reflect progress in Bolivia.

## Integration

The disease modeling has been informed by, as well as shaped, our integrated human practices. Designing this model prompted us to establish lines of communication with experts in Chagas disease (including Dr. Cristina Alonso-Vega and Dr. Yves Carlier) and mathematical biology (Dr. Michael Bonsall). It also played a role in our decision to design a public health poster for individuals living with Chagas disease – most of the mothers of infected children may not be able to obtain treatment and will thus remain in the chronically infected group. Public health campaigns are invaluable, as they aim to educate these women, as well as other infected people, about what to expect when living with Chagas disease and how to protect others from infection.