Applying Individual-based Modeling Approach on Neisseria Gonorrhoeae

Applying Individual-based Modeling Approach on Neisseria

Gonorrhoeae

A Case Study Evaluating Effects of Contact Tracing

By Oda Melina S. Joramo Supervisor: Hans Olav Melberg

Thesis submitted as a part of the Master of Philosophy degree in Health Economics, Policy and Management

Department of Health Management and Health Economics

The Faculty of Medicine

June 2021

Applying Individual-based Modeling Approach on Neisseria

Gonorrhoeae

A Case Study Evaluating Effects of Contact Tracing

By Oda Melina S. Joramo

Supervisor: Hans Olav Melberg

© 2021 By Oda Melina S. Joramo Supervisor: Hans Olav Melberg

Applying Individual-based Modeling Approach onNeisseria Gonorrhoeae http://www.duo.uio.no/

Acknowledgements

First and foremost, I would like to thank my supervisor Hans Olav Melberg. Thanks for your guidance, constant availability, patience, and encouragement. Without his supervision, this thesis would most certainly be of lower quality. Thanks to my mom for sending me flowers and always being there. A big thanks to Ulrikke Nordseth for helping me with the model design and gift card on massage. A special thanks to Martine Kopstad Floeng for commenting on drafts and being vital mental support.

Last but not least, a special thanks to Even Litleskare for bearing out with me and for being a rock when spirits are low.

Abstract

Infection caused by Neisseria Gonorrhea is increasing, mainly among heterosexuals and women. In addition, antibiotic-resistant strains of the bacteria are emerging and rising in Norway. If left untreated, the infection can cause severe damage to those infected, especially women. The aspect of complication caused by gonorrhea has been ignored in the transmission modeling literature of gonorrhea.

This thesis: Applying Individual-based Modeling Approach on Neisseria Gonorrhea: A Case study Evaluating the Effects of Contact Tracing aims to create a novel model and apply this model to a case study exploring the effect of contact tracing on the spread of gonorrhea. The thesis is a modeling experiment incorporating complications into an individual- based SIS framework.

The simulation shows that besides reducing the overall infection level, contact tracing effectively reduces complications. Even if contact tracing does not eradicate the disease, it significantly reduces the number of complications. These results illustrate that incorporating complications into the model generates additional effects relevant to an economic evaluation of interventions.

Contents

1 Background 3

1.1 Background . . . 4

1.1.1 Gonorrhea . . . 4

1.2 Literature Review . . . 7

1.2.1 The literature selection process . . . 7

1.2.2 Review . . . 8

2 Methodology 14 2.1 Methodology . . . 15

2.1.1 Theoretical Framework . . . 15

2.1.2 The Model . . . 16

2.2 Result . . . 26

2.2.1 Result from Model Simulation . . . 26

2.2.2 Parameter Change . . . 30

3 Discussion 34 3.1 Discussion . . . 35

3.1.1 The results . . . 35

3.1.2 Limitations . . . 38

3.1.3 Model Validation . . . 39

3.1.4 Concluding Remarks . . . 40

List of Figures

1.1 Development gonorrhea . . . 4

1.2 Development by Gender . . . 4

2.1 Gonorrhoea Transmission Model Male . . . 16

2.2 Gonorrhoea Transmission Model Female . . . 17

2.3 Total number of Infections . . . 26

2.4 Effect on total symptomatic cases . . . 27

2.5 Effect on total asymptomatic cases . . . 28

2.6 Effect on complications . . . 29

2.7 Simulations Baseline . . . 31

2.8 Average Infections differentβg . . . 32

2.9 Effect of removing the core group . . . 33

List of Tables

2.1 Distribution stable contacts . . . 19

2.2 Distribution Casual contacts . . . 19

2.3 Sexual acts per partnerships . . . 19

2.4 Per act probability of infection . . . 20

2.5 Health state criteria . . . 20

2.6 Distribution of States . . . 21

2.7 Male transition criteria . . . 21

2.8 Female transition . . . 22

2.9 Partner change rate - Stable partners . . . 23

2.10 Partner change rate - Casual partners . . . 23

2.11 Core group . . . 24

2.12 Contact Tracing criteria . . . 24

2.13 Total number of Infections . . . 29

2.14 Incremental Effect . . . 30

2.15 Incremental Effect, percentage-point . . . 30

2.16 Parameter change . . . 32

Introduction

Worldwide, gonorrhea is the second most commonly reported sexually transmittable infection (STI) [21]. In Norway it is categorized as a public health threat [8] and the spread of the infection has steadily increased over the last ten years. Historically gonorrhea has been chiefly reported among men who have sex with men (MSM). Lately, the incidences are increasing more among the heterosexual population than MSM [5]. As a direct consequence of this, an increasing number of women are being infected, and they have a greater risk of developing complications [25].

At the same time, the sexually transmitted bacteriumNeisseria gonorrhoeae (NG) has proven to be highly adaptive to antibiotics, and without effective ways to treat gonorrhea, the health consequences can be severe.

The main objective of this thesis is to create a novel model and apply this model to a case study exploring the effect of contact tracing on the spread of gonorrhea.

This thesis applies an individual-based model (IBM) and incorporates complications as a health state into the SIS (Susceptible - Infected - Susceptible) framework. The agents’ probability of being infected is determined by the disease prevalence, each agent’s number of contacts, and a random variable. The agents’ contact pattern change during the cycles and varies between stable and/or casual partnerships. The probability of developing complications depends on the time an agent stays in an infectious state. Complications have never before been explicitly incorporated into the SIS modeling framework. This thesis explores new aspects of gonorrhea infection and its spread and evaluates contact tracings effect.

The results suggest that contact tracing scenarios that cannot eradicate

causes. This, in turn, implies that contact tracing has positive effects besides decreasing the overall infection level in the population.

The thesis is divided into three main chapters, Background, Method- ology, and Discussion. The Background consists of general information about the disease and the disease trends in Norway. The literature review and a theme summary are also included in the Background chapter. In the Methodology part, the Model, Method, and programming descriptions are presented. The main result from the analysis will also be described here.

The result and the model are assessed in the last chapter, Discussion.

Chapter 1

Background

1.1 Background

In this section, the development of gonorrhea are presented. Also, a description of the natural history of the disease and a short description of Norwegians sexual behaviors are discussed. The information here presents some of the background information applied in the model.

1.1.1 Gonorrhea

Over the last decade there has been a steady increase in the reported cases of gonorrhea in Norway (see figure 1.1). The increase is greatest among the heterosexual population, even though the level of infections is highest among men who have sex with men (MSM) [5]. The number of infected women has increased by tenfold during the last ten years (see figure 1.2).

The situation in Norway follows an international trend where gonorrhea is increasing in numerous western countries [5].

Figure 1.1:Development gonorrhea Figure 1.2:Development by Gender

Alongside the increase of gonorrhea cases, the bacteria has proven to be highly adaptive to antibiotics. Worldwide the NG bacteria has developed resistance to all antibiotic classes used for treatment, and even multiresistant strains have emerged. In Norway, 14.3 percent of NG has developed resistance to azithromycin, and 55.6 percent was resistant to ciprofloxacin in 2018 [4]. The increased antibiotic resistance has forced the Norwegian health care system to apply the last effective antibiotics

1 available as their first-line treatment. Without effective ways to treat

1Third generation cephalosporin

gonorrhea, the health consequences can be severe.

The NG bacteria infect warm, moist areas of the body, including the:

urethra, eyes, throat, vagina, anus, and female reproductive tract. NG cannot survive outside the host. Therefore the transmission of the disease relies on a sexual network to spread the pathogen. NG attaches to the sperm and is easily transmitted from men to their partners. How the transmission from women to men is maintained is less apparent [25]. The transmission probability of unprotected sexual acts is assumed to be 20-30 percent for males, and 50-70 percent for females [9].

The symptoms usually occur seven to 14 days after exposure. How- ever, it can take up to six months before symptoms appear, and some indi- viduals never develop symptoms. The asymptomatic individuals are still contagious. Women are more likely to be asymptomatic than males. It is assumed that as many as 50 percent of females being infected never de- velop symptoms, while this is only true for approximately 10 percent of infected males [8].

Since women are more prone to be asymptomatic, they are at greater risk of long-term complications from untreated infections. If left untreated, gonorrhea can ascend the female reproductive tract and involve the uterus, fallopian tubes, and ovaries. This condition is called pelvic inflammatory disease (PID) and can cause severe and chronic pain and cause irreversible damage to the female reproductive organs [25].

Women may also develop blocking or scarring of the fallopian tubes, which can prevent future pregnancy or cause ectopic pregnancy. The gonorrhea infection can pass to a newborn infant during pregnancy and delivery. In severe cases, this can cause blindness. Infections due to gonorrhea lead to a higher risk of hospitalization and are assumed to be more clinically severe than infections caused by other sexually transmitted diseases [29]. In 2019 there were reports of at least 13 Norwegian women being hospitalized due to PID caused by gonorrhea [5]. Worldwide it is a significant cause of disability among young women [29].

Men may experience scarring of the urethra, and the infection can develop a painful abscess in the interior of the penis. The condition can reduce fertility or cause sterility [28]. When gonorrhea infection spreads

to the bloodstream, both men and women can experience arthritis, heart valve damage, or inflammation of the brain or spinal cord lining. These are rare but serious conditions [14].

Since the disease transmission relies on sexual activity, sexual behavior among Norwegians plays a vital role in transmitting the disease. The infection is highest among males and the younger age groups. In 2019, the median age for infection among men with sex with men was 33 years old, while among heterosexual infected men, the median age was 28. For women, the median age was 23 years old. For both men and women, the median age for infection has decreased [5]. According to the average percentage of infected from 2005 - 2019, most of the infections occur in the age group 20-29 years old, followed by 30-39 years [24]. According to data on sexual habits in Norway, most individuals have had relatively few sexual partners, while the minority have several sexual partners. The sexual debut age is decreasing, and sexual experience with the same sex has increased. All over, females have fewer sexual partners than men throughout their life, while MSM has most sexual partners [10].

1.2 Literature Review

This section describes the process of finding and selecting relevant literature, followed by a literature review. At the end of this section, a short repetition and deception of the thesis theme are presented.

1.2.1 The literature selection process

For this literature review, PubMed was used as a primary source for literature. The results formathematical modeling gonorrhea resulted in 864 articles. To narrow down the results, specific search words relevant to the topic were added. This resulted in the following search worlds, restrictions, and their result:

Search word Result

1 mathematical modeling gonorrhea contact tracing 17

2 agent-based simulation gonorrhea 19

3 modeling antibiotic resistant gonorrhea economic evaluation 7 4 modeling antibiotic resistant gonorrhea health evaluation 25

5 cost-effectiveness agent-based gonorrhea 1

To identify those articles most relevant to the thesis, I read through the abstracts of the selected papers. The first selection was mainly based on excluding articles explaining microbiological trials on humans, animals and articles focusing on vaccine development and new drugs or methods to cure gonorrhea. Also, studies in other languages were excluded. Some pieces were not available in PubMed, even though their abstracts were included. If the articles also were unavailable in Google Scholar they were left out of the review. After reading the remaining papers, only modeling studies are included. Articles focusing on estimating specific parameters, literature reviews, and modeling discussion papers were left out. As a result, 18 articles are discussed in the literature review.

1.2.2 Review

All disease modeling studies require a framework to assign the agents to a health state. The natural history of the disease determines the framework. The SIS framework is the most commonly used for infectious diseases where the infected recovers without developing immunity.

Unlike noncommunicable diseases, infectious disease models require a simulation of how the disease spreads from one individual to another. For sexually transmitted diseases, this means the model must incorporate the agents’ sexual contact patterns. The modeling approach can vary from the simpler deterministic compartmental models to complex agent-based network models.

The deterministic compartmental models often use differential equa- tions to describe movement between health states as a stock and flow model. These models are well suited to model high-level system behavior in large populations [30]. In the modeling literature concerning gonorrhea, the scholars often separate the population into one or several subgroups, where sexual mixing between the subgroups depends on mixing rates.

The approach was introduced by Hethcote and Yorke [15] in 1978, who presented the concept of core-²and noncore groups to model the transmis- sion of gonorrhea [15]. The core group, or the sexually active subgroups, are assumed to be one of the driving forces in keeping the infections at an endemic level. It is also believed that mixing between the core and noncore groups greatly affects the transmission of the disease. Individuals within any pair of population subgroups mix randomly, meaning that each agent in a subgroup has an equal chance of contacting every other in other sub- groups. This implies that every person in the same subgroup will have an equal chance of being infected. Later Garnett and Anderson [12] assessed the significance of sexual mixing between core and noncore groups and its effect on the transmission of gonorrhea. Their results confirm the be- lief that high levels of within-group transmission ensure the community’s disease persistence. The compartmental transmission model allows the researchers to examine the effect of interventions targeting specific core

2The term core group is repeatedly mentioned throughout the literature and refers to a specific group of individuals more sexually active than the rest of the population

groups, like men who have sex with men (MSM). Chan, McCabe and Fis- man [3] developed disease transmission models to evaluate how treatment of core groups would affect the spread of the disease. They found that a fo- cus on the core groups would cause a quicker rebound to a pre-treatment baseline in the presence of antimicrobial resistance. These findings were challenged by Trecker et al. [31] who found that prevalence was lowest when the high-risk group was treated [31]. Fingerhuth et al.’s study ex- amined the rate at which antibiotic-resistant gonorrhea spread in two host populations, heterosexual men and men who have sex with men (MSM).

This study indicates that the difference in the host population’s treatment rate, rather than the difference in sexual partners, explains the differential spread of resistance [7].

Compared to other methods, the compartmental modeling approach is relatively easy to calibrate and fit data [23], and it is widely used to examine different topics. Usually, the method does not specify micro- level behaviors of individuals like interaction and changes over time.

Therefore, this approach is not always the most suitable when individual characteristics play an essential part in spreading the disease [30] [35].

In attempts to better capture human interaction features, later studies have applied network models. Network models can integrate several aspects of human behavior, like allowing contact links to be permanent or semi-permanent and allow the infection to be restricted to a smaller subset of individuals. [35]. The most frequent network model approach applied in the gonorrhea modeling literature is scale-free network models.

The scale-free network model connects individuals with a probability that depends upon the number of partners. They allow a few individuals to have many sexual partners, while most only have a few. This method then allows for highly connected individuals and the creation of a core group [35].

In their study Vajdi et al. [33] explores, using a scale-free model, what role casual partnerships play in the spread of sexually transmittable diseases. Sine casual partners are less likely to be notified about the disease, they play a vital role in maintaining the disease in a population.

The result of this analysis shows that casual sexual behavior and short

partnerships make the population vulnerable to the infection spreading [33]. Whittles, White and Didelot creates a scale-free network model to investigate how a model incorporating a stochastic process of dynamic partnerships formation and dissolution produces more realistic results than static network models or homogeneous mixing models [37]. Using a different approach than Fingerhuth et al. [7] they also find that more sexual partners do not seem to be the main driving force of the spread of antibiotic-resistant gonorrhea [37]. These models can be used to analyse how and why networks change over time and test structural and social influences on the development of health behaviours and outcomes.

However, the method is not well suited to consider higher-level system properties [30].

The most advanced modeling approach, individual-based modeling, complements and extends the aforementioned methods. These models explicitly imitate the entire network of agents and their partnerships, as well as incorporating network dynamics [35] [30]. In an individual-based model, the agents can take on a finite collection of states. The state of an agent at a given point in time is determined by a set of rules that describes the agent’s interaction with other agents. The rules can be either stochastic or deterministic. Additionally, the agent’s state depends on its previous state and the state of other agents with whom they interact [20].

One of the early studies using an individual-based model Kretzschmar et al. [19] simulated the spread of gonorrhea applying a model based on a stochastic pair formation and separation process to describe the underlying structure of sexual contact patterns. It was modeled in an age-structured heterosexual population with a highly sexually active core group. The study aimed to examine different prevention strategies and their effects. Their result suggests that contact tracing is highly effect- ive, and screening should be targeted at the core group. Also, consistent condom use can contribute substantially to the prevention of STDs [19].

In their study Ghani, Swinton and Garnett [13] explores how patterns of sexual mixing and network structures affect the transmission of go- norrhea. They apply an individual-based stochastic simulation of sexual partnerships and disease transmission. The model explicitly generates

different partnership networks from other underlying processes and ex- amines what role these structures play in the transmission dynamics of STDs. Their result suggests that the probability of infections highly de- pends upon the proportion of non-monogamous pairs. At the same time, the prevalence of the disease is sensitive to the mixing patterns of the high activity group [13].

Due to its flexibility, the individual-based approach is widely used in the modeling of complex ecological systems, and lately, the models are becoming more common in epidemiology [35]. Like Hui et al.[16] who used an individual-based model to investigate the potential for imported NG strains to persist in the MSM population in Australia. Vegvari et al. [34] and Zienkiewicz et al. [39] used the approach to explore how Antimicrobial Resistant (AMR) point-of-care tests (POCTS) can affect the usefulness of existing antibiotics. While Weiss et al. [36] studies the effects of expedited partner (EPT) therapy in MSM population in the USA.

Compared to the other models, the individual-based modeling ap- proach can incorporate population heterogeneity at a remarkably detailed level and assess complex interventions. This makes the approach ex- tremely flexible, and IMB can be applied to answer a wide variety of re- search questions. However, it requires substantial computational time and skill and reliable data on a highly detailed level [23].

Among the studies included in this review, most articles apply some version of the SIS framework to simulate the transition between different states. Many of the studies separate the infection states into symptomatic and asymptomatic infections [19, 32, 37, 38]. At the same time, others separate infectious states according to antibiotic-resistant bacteria strains.

How the model is framed depends on the purpose of the study. A majority of the studies focus the analysis on men who have sex with men since the prevalence of gonorrhea and other sexually transmitted diseases has been higher in these groups. However, some studies include both genders, or differentiate between sexual activity rater than gender [3, 7, 12, 13, 19, 32, 33]. None of the aforementioned disease spread models include complications directly into their model. There are studies specifically looking into screenings strategies, and their effect on the probability of

developing Pelvic Inflammatory Disease (PID) from sexually transmitted diseases [29, 26]. Nevertheless, these studies are not concerned with explicitly incorporating PID into a disease spread model.

When modeling sexually transmittable diseases, there seems to be no consensus among scholars on the best approach. In the compartmental transmission models or Markov models, parameters are often constant over time. The heterogeneous sexual patterns are included by separating the population into the core- and non-core groups or subgroups that mix according to specific mixing rates. These models assume that individuals in the same subgroup have the same probability of infection, which is somewhat unrealistic since the likelihood of being infected will be zero for most individuals. However, scholars widely apply this method since it is relatively straightforward to calibrate, is relatively easy to fit data, and is mathematically tractable. The more complex network models can capture specific characteristics of sexual patterns but are challenging to implement. The most ambitious models, the individual-based models (IBM), allow a realistic partnership network to be formed. By explicitly modeling the entire network of individuals and their partnerships, these models can be used to generate different networks, depending on the chosen partnership formation and dissolution rules [35]. The IBM model also allows partnership changes and infection transitions to be stochastic.

Unlike individuals in the compartmental model, the agents are unique and autonomous entities that can locally interact with other individuals [27].

It also enables only a few individuals to be infected, while the majority stay uninfected. However, the modeling approach demands considerable computation skills and time. In addition to this, the realistic partnership network is created at the cost of mathematical tractability.

Theme

As the spread of gonorrhea rises and antibiotic-resistant strains are emerging, the infection will continue to cause severe consequences for young women in Norway (and the rest of the world). The complications caused by gonorrhea come as a direct consequence of women being infected without receiving treatment, most likely due to asymptomatic

infection. The complication aspect of the disease has been ignored in the transmission modeling literature of gonorrhea. This thesis attempts to incorporate complications into an individual-based modeling SIS framework and apply this model to examine the effect of contact tracing.

Chapter 2

Methodology

2.1 Methodology

When individual interactions give rise to population patterns of infec- tion disease incidences and persistence, the individual-based modeling approach is a natural fit. Individual-based models (IBM) provide fur- ther insight into infectious processes in real-world settings. The method is a helpful tool to evaluate infection control policies for decision makers because it allows the researcher to examine the effects of different inter- ventions in virtual experiments. The models will enable the researcher to calibrate various what-if-scenarios and investigate how different interven- tions perform under other potential circumstances. However, the unlim- ited possibilities the IBM approach offers can make the models unneces- sarily complex for its purpose [30, 20, 1].

With the aim of incorporating complications into the model, this thesis creates a novel individual-based model (IBM) to examine contact tracings effect on the spread of gonorrhea. A modified SIS framework allows the creation of several infectious states the agents can transit through. While, the IBM allows agents to transit between states depending on time spent in a state and a random variable. In addition, the IBM approach generates realistic partnership formation and dissolution patterns without building the entire network of individuals and their partnerships. In the succeeding section the details about the the model framework is outlined.

2.1.1 Theoretical Framework

The disease dynamics follow the the SIS model structure to account for recovery without immunity [35]. This model’s structure is the basis for the model created in the thesis. In its simplest form, the agent’s movement between states in the SIS model is determined by the following difference equations:

S_t+1=St −_λt·St+_δ·It (2.1)

I_t+1= It+_λt·St−_δ·It (2.2)

As the equation illustrates, the number of individuals in a state at time t+1 is determined by how many were in the state the previous period t, plus those entering the state, minus those leaving. The movement between the state is determined by the force of infectionλt and the recovery rateδ.

Individual-based modeling extensions to the SIS framework are often used to introduce individual heterogeneity and more complex interaction patterns into the model. It is a computational approach in which agents with a specified set of characteristics interact with each other and their environment according to predefined rules. A defining feature of IMB is that it allows the emergence of population-level phenomena greater than or different from what would be expected based on the only aggregate of individual behaviors [30, 35, 20]. This approach will be used to simulate disease spread and movement between health states.

2.1.2 The Model

In the modified SIS model created for the male agents in this thesis, the infectious state is divided into Symptomatic Infection, Asymptomatic Infection. The Male model structure in figure 2.1 is similar to models used in other studies [38, 32, 37].

Figure 2.1:Gonorrhoea Transmission Model Male

In the female version of the model and additional state, Complications, is also included (see figure??).

Figure 2.2:Gonorrhoea Transmission Model Female

The agents can move between states according to the direction of the arrows. The movement between the states will in this model be determined by an individual random variable ξ_i ¹, individual force on infectionsλⁱ_tand the time spent in an infectious stateT(I).

The states

The main reason for incorporating Asymptomatic Infection is that these will contribute to the spread of the disease to a greater degree than Symp- tomatic infections. Both genders are at risk of developing asymptomatic infections. However, women are at greater risk of being asymptomatic, so women are more prone to developing complications like Pelvic Inflam- matory disease. In addition to this, women are at greater risk of develop- ing long-term complications after infection [29, 25]. Therefore the Female model structure, figure??, includes an additional infection state, Complic- ations, to capture these cases.

1A random number between 0 and 1

Probability of infection

In the model, the traditional force of infection λt is replaced by an individual force of infectionλⁱ_t. Each agent’s probability of being infected will depend on the unique random variable ξⁱ, and the individual force of infection λⁱ_t. The individual force of the infection, λⁱ_t, is given by the equation 2.3, and is determined by the agents number of casual and/or stable contacts, the agents per contact transmission probabilityβ^jg_t and the prevalence of the infection _N^It.² It is the sum of all the individuals in an infection state at timet.

λⁱ_t =

∑

c_jβ^jg_t It

N (2.3)

The agents per contact transmission β^jg_t depend on a gender specific transmission probability per sexual act βg (see table 2.4), the average number of sexual acts per contact (α_j). It is given by equation 2.4:

β_t^jg =

∑

1−(1−_βg)^αj (2.4) Because monogamous partnerships are unlikely to spread the disease, frequency dependent force of infections tends to overestimate the contri- bution of long-term partnerships in the spread of the disease [18]. There- fore the average number of sexual acts per contact (α_j) will differ between stable and casual partnerships (see table 2.3).

Individuals are infected in the simulation if they are susceptible, and their random variableξi is less thanλⁱ_t. The chance of being infected will to some degree be stochastic, but the possibility ofλⁱ_t being greater thanξ will increase by an increased number of partners.

Model Parameters

At the start of each simulation each agent in the model will be assigned a number of stable and casual contacts, the number of casual contacts is based on casual partners from Weiss et al.’s study[36]. The distribution of steady partnerships is based on a Norwegian sexual behavior report

2j=stable(s),casual(c)

[10], and assignment to stable partnerships are based on the share of individuals who reported living with a partner (see table 2.1).

Stable 0 1

Both Genders 0.32 0.68

Table 2.1:Distribution stable contacts

The distribution of casual contacts is based on lifetime sexual partners data from the same report [10]. The share of males and females reported having 0-1, 2-5, and 6-11+ sexual partners [10] and own calculations is the basis for those assigned to respectably 0, 1, and 2 casual contacts (see table 2.2).

Casual 0 1 2

Male 0.23 0.47 0.30 Female 0.28 0.51 0.21

Table 2.2:Distribution Casual contacts

To address the problem of overestimating long-term relationships contribution to the spread of the disease, αj, will differ for steady and casual partnerships. In literature, the average number of sexual acts per partnership is estimated to be three [11]. Data shows that an estimated 40 percent of every partner is unfaithful in their steady partnership [2].

Therefore α_stable will be a weighted average based on the percentage of infidelity and estimated sexual acts. Where the sexual acts in 60 percent of the cases are set to zero, three otherwise (see table 2.3).

j Stable Casual

αj 1.2 3

Table 2.3:Sexual acts per partnerships

In the literature, the per act probability of infectionβgranges from 0.2 to 0.3 for the male agent, and between 0.5 to 0.7.Each simulation the model draws from a uniform distribution (see table 2.4).

Male Female

βg [0.2 - 0.3] [0.5 - 0.7] Uniform distribution

Table 2.4:Per act probability of infection

Symptomatic or Asymptomatic

Once infected, the agents must either transit to the symptomatic infection state or the asymptomatic infection state. Because female agents can develop symptoms after being asymptomatic, more agents are assigned to this state than expected after some cycles. The probability of being asymptomatic differ across the genders; therefore, the criteria differ. The assignment to the states depends on the criteria presented in table 2.5.

INFECTION CRITERIA STATUS CRITERIA INFECTION STATUS Male Agents

ξ_i<λⁱ_t ξ_i ≥ ₁₀^λit Symptomatic Infection ξi<λⁱ_t ξi < ^λ₁₀^it Asymptomatic infection Female Agents

ξi<λⁱ_t ξi ≥ ^λ₃^it Asymptomatic infection ξi<λⁱ_t ξi < ^λ₃^it Symptomatic Infection

Table 2.5:Health state criteria

A majority of the male agent will develop symptoms. The assignment is determined by dividing the λⁱ_t by ten, generating a relatively small number, allowing only those agents with a sufficiently small ξ_i to be assigned to the asymptomatic state. A larger proportion of the female agents will be asymptomatic. Therefore λⁱ_t is divided by three, and a bigger proportion of females will be in the asymptomatic state. Since some asymptomatic infections develop symptoms during the simulation, the proportion assigned to asymptomatic infections will be higher than those assigned to the symptomatic state when first infected. The assignment will to a great extent, be stochastic dependent on the agent’s random variable ξiand theirλⁱ_t.

Movement between states

Before each simulation starts, all the agents in the model will be assigned to a health state based on the distribution in table 2.6 based in data from [24]. In addition to this the agents will be randomly assigned the time spent in that state between, ranging from 0 to 6 cycles.

State Susceptible Symptomatic I Asymptomatic I Prb.Dist. 0.999 0.00075 0.00025

Table 2.6:Distribution of States

Once infected and assigned to either symptomatic or asymptomatic infection the agents will follow a semi-stochastic process to recovery or developing complications. The transition probability will mainly be determined by three factors, the agent’s gender, the time they have spent in an infectious state and their random variable ξi. The transition from one state to another will in this model be determined by the time spent in that state T(I), and ξ_i. To transit, the T(I) must exceed the different criteria illustrated in table 2.7 and 2.8. Otherwise, the individuals will stay in the same state until they satisfy a criteria and can transit. The longer the agents remain in one state, the higher the probability of satisfying one or more criteria.

From state To state Criteria Symptomatic Susceptible T(I^s)> 3∗ξi

Asymptomatic Symptomatic T(I^a)_{> 7}∗_ξi Asymptomatic Susceptible T(I^a)> 13∗ξ_i

Table 2.7:Male transition criteria

Once in the symptomatic state, the male agent will seek treatment with some delay. When individuals seek treatment, they will eventually re- cover. To capture that not all individuals will seek treatment immediately after symptoms appear, this model allows the recovery process to be some- what random. All males with a symptomatic infection will recover after a

in that state until they satisfy a criteria and transit. The criteria for devel- oping symptoms is lower than moving to susceptible because the model assumes the agents are more likely to develop symptoms and seek treat- ment than the disease recovering naturally. The logic for females’ trans- ition is similar but with an extra state to possibly transit through.

From state To State Criteria Symptomatic Susceptible T(I^s)> 3∗ξ_i Asymptomatic Symptomatic T(I^a)> 7∗ξi

Symptomatic Complications T(I^s)> 13∗ξ&ξi<0.2 &T(I^s)> 2 Asymptomatic Complications T(I^a)> 13∗_ξi &ξi <0.2

Asymptomatic Susceptible T(I^a)> 13∗ξ_i &ξ_i ≥0.2 Complications Susceptible T(I^a)> 0

Table 2.8:Female transition

The female agents’ who develop symptoms will seek treatment with some delay, depending onξi. They are still at risk of developing complic- ations even though it is more unlikely when developing symptoms early.

The asymptomatic infected can, like the male agent, develop symptoms and recover naturally, in addition to developing complications if left un- treated. After three cycles of being infected with symptoms, all the agents will either develop complications or be susceptible again. The models as- sume that the asymptomatic agents will develop symptoms, recover nat- urally, or develop complications after a maximum of seven cycles. The chance of developing symptoms or recovering without treatment is almost equally likely, depending on ξ_i. Once an agent enters the complications state, they will always recover after one cycle.

The transition from one state to another is set up in this way to be able to capture that how long an individual stays in one state affects the outcomes of the disease. The reason is that the model wants to incorporate that women going for an extended period without receiving treatment can develop complications.

Partner-change

During each cycle, the agents can change their number of stable and casual contacts. Casual partnerships are categorized as having higher dissolution rates than stable partnerships. According to data, males have more casual partners than females [10, 33], therefore the model allows males to increase their number of casual partners to three during the simulation. Partner change in the model will happen according to the distribution in table 2.9 and 2.10. The distribution in the tables (2.9 and 2.10) are based on data from the literature [19, 17] and transformed to fit the cycle length of one month.

Male and Female

-1 0 +1

0 - 0.83 0.17 1 0.012 0.988 -

Table 2.9:Partner change rate - Stable partners

Male Female

-1 0 +1 -1 0 +1

0 - 0.83 0.17 - 0.83 0.17 1 0.003 0.827 0.17 0.003 0.827 0.17 2 0.006 0.824 0.017 0.006 0.994 -

3 0.009 0.991 - - - -

Table 2.10:Partner change rate - Casual partners

According to Norwegian data on gonorrhea infections [24] the infec- tion rates are highest among the age group 20-29 and more males than females get infected. This is because this age group is the most sexually active [5]. Therefore, a small proportion of the agents in this age group are assigned to a more sexually active core group (see table??). In this core- group the number of contacts will randomly change between 0 - 3 contacts each cycle.

Age 20-29 Male Female Core-group 0.0005 0.00032

Table 2.11:Core group

Contact Tracing

The main purpose of contact tracing is to detect those who otherwise would not seek treatment. The model will simulate contact tracing by assuming that asymptomatic agents recover earlier due to treatment.

Those individuals developing symptoms will seek treatment at the same rate as before. Therefore the model does not change the transition probabilities for these states (see Table??).

Asymptomatic to Susceptible Critera Contact Tracing 1 T(I^a)> 5∗ξ_i

Conact Tracing 2 T(I^a)> 3∗ξi Table 2.12:Contact Tracing criteria

Two contact tracing scenarios are presented in this thesis. The difference determines how long the agents can stay asymptomatic without being discovered and treated. In the first scenario, Contact Tracing 1, the longest the agent is allowed to stay asymptomatic is five cycles. While Contact Tracing 2, allows the agent to be asymptomatic for maximum three cycles.

Assumptions

The model simulates condom-free sexual contact between male and female agents. There is no distinction between anatomical sites of the infection, and because of cycle length, the incubation period is ignored in the calculations. The time horizon is set to 36 months, where each cycle represents one month. The population in this model consists of N

= 100.000 male and female agents in the ages 15-59 years old. Due to the short time horizon, the population is closed and stable, meaning no agents

will enter or leave the population. All agents will recover after receiving treatment.

Simulation in Python

The Python programming software was applied to execute the simulation of the model described. For each cycle in the simulation, the model will infect new individuals (see table 2.5, 2.7 and 2.8), update the infected agents’ state and change the agents’ number of contacts (see table 2.10 and 2.9).

2.2 Result

In this section, the result from the model simulations will be presented.

Since, the model includes stochastic elements, the result from one simulation will differ some from the previous simulation. Comparing two random simulations with and without the intervention will not show the correct effect. Therefore, the result presented in the following section will all be the average of ten simulations. Also, the results from changing the transmission probability per actβgand excluding the core group from the model are presented.

2.2.1 Result from Model Simulation

Figures 2.3, 2.4, 2.5, and 2.6 show the average number of infections from ten simulations, with and without contact tracing. The Baselines in the figure represent the spread of the disease without any interventions. The figures present two different contact tracing scenarios, Contact Tracing 1 and Contact Tracing 2 (see table 2.12 Section 2.1). Figure 2.3 illustrates the effect of contact tracing on the overall spread of the infection in the model.

While figures 2.4, 2.5 and 2.6, shows the effect of contact tracing on the different infectious states.

Figure 2.3:Total number of Infections

Figure 2.3 illustrates the total number of infections under three different scenarios, a do-nothing scenario (Baseline) and two contacts tracing scenarios (Contact Tracing 1 and Contact Tracing 2). The Baseline quickly stabilizes at an endemic level where the number of infections per cycle lies between 200 to 225 cases. The Contact Tracing 1 line stabilizes at a lower level than the Baseline, with 150 to 175 Infections per cycle.

Under the second contact tracing scenario, the disease has a declining trend, throughout the time-horizon.

Figure 2.4:Effect on total symptomatic cases

In figure 2.4, the number of symptomatic infections under the same three scenarios is presented. The overall trend is similar to the trends shows in figure 2.3. According to the Baseline, number of symptomatic infections quickly stabilizes at an endemic level, where the number of infections lies between 110 and 130 each cycle. Under Contact Tracing 1, the number of cases decreases until it stabilizes at a lower level than the Baseline, where the number of Infections lies between 60 and 80 Infections per cycle. While, the infection steadily decreases and doesn’t stabilize during the time horizon, in the Contact Tracing 2 scenario.

Figure 2.5:Effect on total asymptomatic cases

Figure 2.5 shows the development of asymptomatic infections. The Baseline asymptomatic cases stabilize at an endemic level after five cycles, with approximately 70 to 80 asymptomatic infections each cycle.

Implementing contact tracing causes the infections to decrease until cycle 15, where the trend stabilizes. In the first scenario, Contact Tracing 1, the disease stabilizes with 40 to 50 asymptomatic cases per cycle. While in Contact Tracing 2 stabilizes with 20 to 40 asymptomatic cases per cycle.

Figure 2.6:Effect on complications

The last figure illustrating the effect of contact tracing, figure 2.6, shows the number of infections developing complications in the three scenarios.

In the Baseline scenario the level of infection stabilizes and lies between 30 and 45 cases per cycle. After cycle 20 Contact Tracing 1 stabilizes with 20 to 25 number of complication cases per cycle. Compared to Contacts Tracing 1, Contact Tracing 2 stabilizes at a lower level with 10 to 20 complications each cycle.

Figures 2.3, 2.4, 2.5 and 2.6 illustrates that contact tracing decreases the number of infections. For all the states the level of infections is lower when contact tracing is implemented. In table 2.13, the total average number of infections from the figures is presented.

Is Ia Ic I

Baseline 4126 2861 1235 8222 Contact Tracing 1 2610 1726 795 5133 Contact Tracing 2 2075 1503 653 4230

Table 2.13:Total number of Infections

Table 2.13 shows the same as the figures, that overall, implementing

number of avoided infections is given in table 2.14.

Table 2.14 shows the incremental effect in avoided cases going from Baseline to Contact tracing 1 and the incremental effect going from Contact Tracing 1 to Contact Tracing 2. The number of avoided cases decrease with 3091 going from a do-nothing situation to implementing Contact Tracing 1. The total number of avoided cases decreases, with 901 going from Contact Tracing 1 to Contact Tracing 2.

∆Is ∆Ia ∆Ic ∆I

Baseline - - - -

Contact tracing 1 1515 1135 441 3091 Contact tracing 2 536 224 141 901

Table 2.14:Incremental Effect

Table 2.15 shows the change in avoided cases going from baseline to Contact Tracing 1 and going from Contact Tracing 1 to Contact Tracing 2.

Table 2.15 illustrates that the most significant change in avoided cases is obtained going from do-nothin to Contact Tracing 1, reducing the number of infections by 37 percentage points.

∆Is ∆Ia ∆Ic ∆I

Baseline - - - -

Contact tracing 1 0.36 0.39 0.35 0.37 Contact tracing 2 0.25 0.12 0.17 0.17

Table 2.15:Incremental Effect, percentage-point

2.2.2 Parameter Change

Since the model has incorporated stochastic elements, each simulation represents different scenarios and their outcomes. Figure 2.7 illustrate the ten simulations making up the average Baseline variable.

Figure 2.7:Simulations Baseline

Among the simulations, a majority stabilizes at different levels gener- ating an endemic spread of the disease. However, there are a few outliers.

In simulation 2 the disease is spreading while it is eradicated in simulation 3. These diverging results indicate that some mechanisms can significantly change the outcome of the simulation.

The rest of this section will present how different parameters and the core group affect the outcome. Unlike figure 2.7 the result presented in the following section will again be the average of ten simulations.

To examine the effect parameter change has on the outcome, the simulation is ran using different fixed parameter values of the per act probability of being infected βg (see table 2.16). The per act probability will differ from being high for both genders (HH), low for both genders (LL), or a mix between high for and low (FLMH, FHML)³.

3FLMH= Female Low, Male High and FHML = Female High, Male Low

βFemale βMale

HH 0.7 0.3

FLMH 0.5 0.3

FHML 0.7 0.2

LL 0.5 0.2

Table 2.16:Parameter change

Figure2.8 illustrates the effect of fixing the per act probability βg

according to the values in table 2.16.

Figure 2.8:Average Infections differentβg

In figure 2.8, the baseline, FLMH, and FHML scenarios stabilize at different endemic levels. While in the HH scenario, the disease is increasing at a relatively high level. Comparing FLMH and FHML illustrates that a higherβ_male gives a higher endemic level.

Figure 2.9 presented below illustrates the effect of removing the core group from the model.

Figure 2.9:Effect of removing the core group

Removing the core group significantly impacts the outcome of the model. Without the more sexually active core group the number of infections steadily decreases throughout the time horizon.

The results produced by the model demonstrate that without any interventions, the disease establish itself as endemic in the model population. Implementing Contact Tracing 1 reduces the total number of infections, however, the disease stabilizes and stays present among agents in the population. Contact Tracing 2 reduces the number of infections further. However, whether Contact Tracing 2 eradicates the disease is uncertain. Figures 2.8 illustrates how the stochastic variables generates varying outcomes each simulation. Combinations, whereβ_maleandβ_{f emale} are at their highest (HH), enable the disease to spread rapidly. While all other combinations of βg establishes the disease spread as endemic. But, without the core group, the disease struggles to stay endemic within the model population, and is steadily decreasing (see figure 2.9)

Chapter 3

Discussion

3.1 Discussion

This thesis is an experiment in applying a new modeling method to assess the effects of contact tracing of gonorrhea infections. The created model include complications caused by untreated gonorrhea into an individual- based SIS framework. In the upcoming section, the results and the model will be discussed.

3.1.1 The results

Unlike the smooth curved graphs usually presented in the infectious disease literature, the outcome presented here shows a jagged line (see figure 2.3, 2.4, 2.5 and 2.6). The pattern emerging is most likely a result of how the model is calibrated. The transition mechanisms are determined by the time spent in a state (T(I)) and random variables ξ_i. There are no fixed transmission probabilities, like the ones seen in stock and flow models. In addition to this, the model is programmed to have few highly sexually active individuals with a higher probability of becoming infected, which means that the number of susceptible in the models does not represent those agents at risk of becoming infected. Creating individual probabilities of infection, λⁱ_t, that increases with the agents’

number of contacts permits the most sexually active to have a dissimilar probability of infection, while those with no contacts stay uninfected.

Consequently, the highly sexually active agents in the model will quickly become infected, causing the number of infections to rise. The infected agents stay infectious for some cycles before returning to the susceptible pool. Before the agents become susceptible, the number of agents at risk of infection will already be infected, and the number of infections decline.

As the number of new infections rises, the number of potential agents to infect decreases, and the jagged line shown in figure 2.3 is generated.

The baseline scenario in figure 2.3 establishes itself as endemic in the model population, which means that the disease’s reproductive number equals one and that each infected agent passes on the infection to at least one other agent. An endemic disease can stay within a population without increasing or go to extinction. Figure 2.3 shows that in the different contact

tracing regimes disease spread decreases in the model population. The characteristics of disease development are unequal in the two contact tracing scenarios. In the first scenario, Contact Tracing 1, the infection stabilizes at a lower endemic level. The intervention is not efficient in removing the disease from the model population. The second scenario, Contact tracing 2, is much more efficient in decreasing the spread of the disease. It is unclear whether Contact Tracing 2 removes the disease entirely from the model population or if it stabilizes at an even lower endemic level.

The contact tracing scenarios are modeled by assuming that the asymptomatic infections recover more quickly than they would without contact tracing. Implementing contact tracing reduces the incidences of symptomatic as well as the targeted asymptomatic infections (see figure 2.4). The shift in symptomatic cases due to contact tracing indicates that several asymptomatic infections recover before developing symptoms and therefore reduce the number of symptomatic cases as well as the asymptomatic cases.

Tables 2.14 and 2.15 state that the effects of contact tracing on avoided complications are most significant when going from the Baseline (do- nothing) scenario to implementing Contact Tracing 1. The number of avoided complications is reduced by a total 35 percentage points. In comparison, the change in avoided complications going from Contact Tracing 1 to Contacts Tracing 2 was only 17 percentage points. This implies that the effect of contact tracing on complications is most significant when agents seek treatment before staying infected for six months. According to the model simulations, most agents will develop complications between cycles three and six. Therefore the effect of uncovering asymptomatic infections before three cycles is not as significant compared to Contact Tracing 1.

Of the two contact tracing scenarios, the most ambitious, Contact Tracing 2, is the most effective in reducing the overall spread of the disease. It almost brought the disease to extinction and reduced the total number of complications by 582 cases. These results are similar to what Kretzschmar et al. [19] found when assessing different disease spread

interventions.

However, more interestingly, the least effective intervention in redu- cing the disease spread effectively reduced the number of avoided com- plications. This illustrates that by incorporating complications into the model, new outcomes of the interventions can be evaluated. This is useful if decision-makers want interventions to affect other aspects of an infec- tious disease than the disease spread. Including the addition effect dir- ectly into the model is also valuable for cost-effectiveness analysis, where all the effects of interventions are relevant to include.

Parameter Change

Each cycle, the agents in the core group randomly change their number of contacts while those in the noncore group change their number of stable and casual partners according to a probability distribution. In addition, the per act probability βg is drawn from a uniform distribution (see table 2.4). Together the per act probability of infection, the agent’s number of partners, and the prevalence of the infection determine the disease’s infectiousness, λⁱ_t. Different combinations of contact patterns and per act probabilities permits the outcome to differ from one simulation to the next, allowing several disease equilibria to exist (see figure 2.7).

A majority of the combinations establish the disease at an endemic equilibrium in the model population. However, a few outliers indicate that some combinations will dramatically alter the result (see figure 2.7).

This diverging result substantiates the importance of running several simulations to produce the average variables. Comparing two random simulations with and without contact tracing could then be misleading.

Fixing the per act probability and removing the core group enables one to look into which combinations primarily affect the disease’s spread, (see figure 2.8). The result illustrates that when the per act probability of infection for both males and females is high (HH), the infection spreads in the model population. Both FLMH and FHML stabilizes at endemic levels, FLMH at a lower level than FLMH. These results suggest that in High Low combinations of per act probability, the high value affects the disease level most when it belongs to the most sexually active group, in

this case the male agents. When both βg’s are low (LL), the infection is still endemic in the model population, suggesting that even less infectious sexually transmitted diseases will be present in the model population.

These findings suggest that the more sexually active agents and the core group enable the infection to sustain in the model population.

By removing the sexually active core group from the model, the results are greatly altered, causing a steady decrease in infections (see figure 2.9). The importance of a sexually active core group to sustain a sexually transmitted disease within a population is widely documented in the literature [15, 12, 3]. Combined with highβg(HH), the presence of a core group allows the disease to spread rapidly. The increased prevalence will, in this case, contribute even further to the spread. In a scenario with low βg, (LL), combined with a core group, the disease stays endemic; in these cases, the core group is believed to serve as a reservoir of the infection [12]. In these cases, interventions solely targeting core groups rather than the whole population can be effective in reducing a sexually transmitted disease [12, 3, 38].

3.1.2 Limitations

All models have their limitations, and this is no exception. The empirical partnerships data applied in this thesis originate from different countries where the population may have different partnership distributions than the targeted population in the model [30, 6]. The model output is based on simulations of do-nothing scenarios or including contact tracing intervention. In reality, condom use, screening, and other factors will affect the spread of the disease. Therefore, the results are not directly comparable to the real-world data.

The model ignores imported infections affecting disease spread and an intervention’s ability to eradicate the disease. In Norway, approximately 20 percent of all infections in 2019 were imported from abroad [24].

Imported infections can reintroduce a disease into the population even though infection levels within the domestic borders are low. However, few modelers attempt to include imported infections due to the challenges it imposes.

Also, the model ignores the agents’ potential behavioral changes. If infection levels increase and become high, people may become more risk-averse and reduce their sexual partners, or deficient levels of infections might make the people more risk-prone likewise care less about the consequences of having several casual sexual partners [22].

These behavioral changes can be included in the model by allowing the contact patterns to change if infections reach certain levels. To incorporate feedback ability and adaption is particularly important when the circumstances may change behaviors and neutralize the intended positive effect on an intervention [30, 20]. In addition to this behavior can change with age, where younger individuals are more sexually active than the older. It is feasible to create different transition probabilities depending on the agent’s age. However, since detailed data on age-dependent sexual behavior are scarce, it is overlocked in the model.

3.1.3 Model Validation

A valid model is not the same as a perfect model. A model can have different levels of validation for different applications [6]. Therefore validity is not a measure of how perfect the outcome of a model is but how well it answers the questions asked. This thesis aimed to create a model including complications explicitly into an individual-based model framework and apply this model to examine the effects of contacts tracing.

By including complications, the model generates different outcomes than other disease spread models. This makes it possible to evaluate if an intervention affects more than the spread of the disease and allows decision-makers to implement interventions with new targets. Letting time spent in an infectious state determine the transition probabilities allows the researcher to examine how the timing of the intervention affects the outcome. The simulations’ result shows that incorporating complications into the model gave new insight into the effect of contact tracing.

In addition to this, the model successfully incorporates that the pre- valence of sexually transmitted diseases strongly depends on casual part-

effects of steady partnerships’ role in spreading the diseases [18]. The cre- ation of separate per partner transmission probabilities β^jg_t for stable and casual relationships makes it less likely for agents engaged in only stable relationships to be infected, and the chance of becoming infected increases with the number of casual partners. Also, the model has incorporated other unspecified influences by including random effects represented by the random variableξi. Combined, these features create a realistic contact pattern among the agents and incorporate that different contacts affect the spread of the infection differently, without generating the entire network of contacts.

This thesis illustrates that other outcomes than disease extinction are potentially relevant when evaluating an infectious disease intervention.

Therefore the created model contributes to the literature by shedding light on new aspects of infectious diseases and examining an alternative modeling approach.

3.1.4 Concluding Remarks

The main objective of this thesis is to create and apply a novel model to examine the effect of contact tracing on the spread ofNeisseria gonorrhoeae (NG). The created individual-based model (IBM) builds on the SIS framework and incorporates several infectious states the agents can transit through. Including complications permits the model to measure different outcomes caused by the disease in addition to the total number of infected.

In addition to this the model generatea realistic partnership formation and dissolution patterns.

The results show that even though an intervention cannot remove the disease from the population, it significantly reduces the number of complications caused by a Neisseria gonorrhoeae infection. Illustrating additional positive effects of contact tracing besides reducing the overall infection level.

A natural expspansion to this thesis would be to examine how targeted contact tracing affects the spread of the disease and the development of complications. Also, increasing the time-horizon and incorporating feedback mechanism could improve the model further.

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