## HIV Model Outline¶

### Goal¶

We will examine policy interventation strategies to address and control the spread of HIV in a sexual network.

### Justification¶

Given the heterogeneity of behavior and network effects of sexually-transmitted diseases, this problem requires ABM and network methodology. Differential equation compartment models will not allow for incomplete or scale-free graph structures or time-varying agent behavior, and so cannot be used.

### Processes and Mechanisms of Interest¶

We are primarily interested in how different ranges of the condom subsidy and condom budget affect the proportion of people that are infected with HIV at each time step. Additionally, we are interested in the rate at which HIV infection could increase given increased condom costs.

### Outline¶

• Implement condom subsidy mechanics
• Implement gossip network to share status
• Vary condom subsidy and gossip network structure
• Observe how number of steps to 50% infected
• Observe the mean and standard deviation of sexual partner degree distribution

To formally describe our model, let's break it down into pieces:

### I. Space¶

In this model, our space will be a two-dimensional (2D) square grid. Each grid cell will contain zero or one people. Edges of the grid will wrap around.

### II. Actors¶

#### A. People¶

In this case, people are the carriers of HIV. We are modeling them as simply as possible, using only the following properties:

Properties

• _isinfected: is the person infected with HIV?
• _condombudget: what is the person's budget for protection?
• _probhookup: what is the probability that a person will want to hookup given contact?

For their step function, agents will perform the following:

• take a unit-distance step in a random direction
• evaluate neighboring cells for any potential people to hookup with
• for each pair, check against _probhookup; if both people sample True, then hookup
• for a hookup, check against _maxbudget and the global condom cost to see if either partner will purchase

#### B. Institution¶

A "public health" institution will manage the level of subsidy for condoms. For now, the subsidy level will be set at $t_0$ and constant over model run.

### III. Initial Conditions¶

#### A. People¶

• People will be randomly distributed throughout the grid by sampling from a uniform discrete distribution with replacement. In the event that an agent has already been placed at the sampled location, we continue drawing a new nandom position until it is unoccupied.
• People will have their prob_hookup randomly initialized to a value from a uniform continuous distribution.
• People will have their condom_budget randomly initialized to a value from a uniform continuous distribution.
• A single person will be selected at random at $t_0$ to have HIV.

#### B. Insitution¶

The public health institution will set a level of subsidy for condoms using a random uniform continuous distribution.

### IV. Model Parameters¶

Based on the description above, we need the following model parameters:

• grid_size: size of the two-dimensional square grid, i.e., dimension length
• num_people: number of persons to create; must be less than ${grid\_size}^2$
• min_subsidy, max_subsidy: the lower and upper bounds for the institution's subsidy
• min_condom_budget, max_condom_budget: the lower and upper bounds for initial conditions of people's condom_budget
• condom_cost: cost of a condom; fixed to $1.0$
• min_prob_hookup, max_prob_hookup: the lower and upper bounds for initial conditions of people's prob_hookup
• prob_transmit, prob_transmit_condom: the probability of transmitting the disease without and with a condom
In :
%matplotlib inline

# Standard imports
import copy
import itertools

# Scientific computing imports
import numpy
import matplotlib.pyplot as plt
import networkx
import pandas
import seaborn; seaborn.set()

# Import widget methods
from IPython.html.widgets import *


## Person Class¶

Below, we will define our person class. This can be broken up as follows:

• constructor: class constructor, which "initializes" or "creates" the person when we call Person(). This is in the __init__ method.
• decide_condom: decide if the person will purchase a condom, i.e., by checking that $condom\_budget >= condom\_cost - subsidy$
• decide_hookup: decide if the person will hookup, i.e., by sampling with $p=prob\_hookup$
In :
class Person(object):
"""
Person class, which encapsulates the entire behavior of a person.
"""

def __init__():
"""
Constructor for Person class.  By default,
* not infected
* will always buy condoms
* will hookup 50% of the time

Note that we must "link" the Person to their "parent" Model object.
"""
# Set model link and ID
self.model = model
self.person_id = person_id

# Set Person parameters.
self.is_infected = is_infected
self.condom_budget = condom_budget
self.prob_hookup = prob_hookup

def decide_condom(self):
"""
Decide if we will use a condom.
"""
if self.condom_budget >= (self.model.condom_cost - self.model.condom_subsidy):
return True
else:
return False

def decide_hookup(self):
"""
Decide if we want to hookup with a potential partner.
"""
if numpy.random.random() <= self.prob_hookup:
return True
else:
return False

def get_position(self):
"""
Return position, calling through model.
"""
return self.model.get_person_position(self.person_id)

def get_neighbors(self):
"""
Return neighbors, calling through model.
"""
return self.model.get_person_neighbors(self.person_id)


## Model Class¶

Below, we will define our model class. This can be broken up as follows:

• constructor: class constructor, which "initializes" or "creates" the model when we call Model(). This is in the __init__ method.
• setup_space: method to create our "space"
• setup_people: method to create our "people"
• setup_institution: method to create our "institution"
• get_neighbors: method to get neighboring agents based on position
• get_person_neighbors: method to get neighboring agents based on agent ID
• get_person_position: method to get position based on agent ID
• move_person: method to move an agent to a new position
• step_move: method to step through agent moves
• step_interact: method to step through agent interaction
• step: main step method to control each time step simulation
In :
class Model(object):
"""
Model class, which encapsulates the entire behavior of a single "run" in our HIV ABM.
"""

def __init__( ):
"""
Class constructor.
"""
# Set our model parameters; this is long but simple!
self.grid_size = grid_size
self.num_people =  num_people
self.min_subsidy = min_subsidy
self.max_subsidy = max_subsidy
self.min_condom_budget = min_condom_budget
self.max_condom_budget = max_condom_budget
self.condom_cost = condom_cost
self.min_prob_hookup = min_prob_hookup
self.max_prob_hookup = max_prob_hookup
self.prob_transmit = prob_transmit
self.prob_transmit_condom = prob_transmit_condom

# Set our state variables
self.t = 0
self.space = numpy.array((0,0))
self.condom_subsidy = 0.0
self.people = []
self.num_interactions = 0
self.num_interactions_condoms = 0
self.num_infected = 0

# Setup our history variables.
self.history_space = []
self.history_space_infected = []
self.history_interactions = []
self.history_num_infected = []
self.history_num_interactions = []
self.history_num_interactions_condoms = []

# Call our setup methods to initialize space, people, and institution.
self.setup_space()
self.setup_people()
self.setup_institution()

def setup_space(self):
"""
Method to setup our space.
"""
# Initialize a space with a NaN's
self.space = numpy.full((self.grid_size, self.grid_size), numpy.nan)

def setup_people(self):
"""
Method to setup our people.
"""

# First, begin by creating all agents without placing them.
for i in xrange(self.num_people):
self.people.append(Person(model=self,
person_id=i,
is_infected=False,
condom_budget=numpy.random.uniform(self.min_condom_budget, self.max_condom_budget),
prob_hookup=numpy.random.uniform(self.min_prob_hookup, self.max_prob_hookup)))

# Second, once created, place them into the space.
for person in self.people:
# Loop until unique
is_occupied = True
while is_occupied:
# Sample location
random_x = numpy.random.randint(0, self.grid_size)
random_y = numpy.random.randint(0, self.grid_size)

# Check if unique
if numpy.isnan(self.space[random_x, random_y]):
is_occupied = False
else:
is_occupied = True

# Now place the person there by setting their ID.
self.space[random_x, random_y] = person.person_id

# Third, pick one person to be infected initially.
random_infected = numpy.random.choice(range(self.num_people))
self.people[random_infected].is_infected = True
self.num_infected += 1

def setup_institution(self):
"""
Method to setup our space.
"""
# Randomly sample a subsidy level
self.condom_subsidy = numpy.random.uniform(self.min_subsidy, self.max_subsidy)

def get_neighborhood(self, x, y, distance=1):
"""
Get a Moore neighborhood of distance from (x, y).
"""
neighbor_pos = [ ( x % self.grid_size, y % self.grid_size)
for x, y in itertools.product(xrange(x-distance, x+distance+1),
xrange(y-distance, y+distance+1))]
return neighbor_pos

def get_neighbors(self, x, y, distance=1):
"""
Get any neighboring persons within distance from (x, y).
"""
neighbor_pos = self.get_neighborhood(x, y, distance)
neighbor_list = []
for pos in neighbor_pos:
# Skip identity
if pos == x and pos == y:
continue

# Check if empty
if not numpy.isnan(self.space[pos, pos]):
neighbor_list.append(int(self.space[pos, pos]))

return neighbor_list

def get_person_position(self, person_id):
"""
Get the position of a person based on their ID.
"""
# Find the value that matches our ID in self.space, then reshape to a 2-element list.
return numpy.reshape(numpy.where(m.space == person_id), (1, 2)).tolist()

def get_person_neighbors(self, person_id, distance=1):
"""
Get the position of a person based on their ID.
"""
# Find the value that matches our ID in self.space, then reshape to a 2-element list.
x, y = self.get_person_position(person_id)
return self.get_neighbors(x, y, distance)

def move_person(self, person_id, x, y):
"""
Move a person to a new (x, y) location.
"""

# Get original
original_position = self.get_person_position(person_id)

# Check target location
if not numpy.isnan(self.space[x, y]):
raise ValueError("Unable to move person {0} to ({1}, {2}) since occupied.".format(person_id, x, y))

# Otherwise, move by emptying and setting.
self.space[original_position, original_position] = numpy.nan
self.space[x, y] = person_id

def step_move(self):
"""
Model step move function, which handles moving agents randomly around.
"""

def step_interact(self):
"""
"Interact" the agents by seeing if they will hookup and spread.
"""

# Get a random order for the agents.
random_order = range(self.num_people)
numpy.random.shuffle(random_order)

# Track which pairs we've tested.  Don't want to "interact" them twice w/in one step.
seen_pairs = []

# Iterate in random order.
for i in random_order:
# Get neighbors
neighbors = self.get_person_neighbors(i)

# Iterate over neighbors
for neighbor in neighbors:
# Check if we've already seen.
a = min(i, neighbor)
b = max(i, neighbor)
if (a, b) not in seen_pairs:
seen_pairs.append((a, b))
else:
continue

# Check if hookup if not seen.
hookup_a = self.people[a].decide_hookup()
hookup_b = self.people[b].decide_hookup()
if hookup_a and hookup_b:
# Hookup going to happen.
self.num_interactions += 1

# Check now for condoms and use resulting rate.
if self.people[a].decide_condom() or self.people[b].decide_condom():
# Using a condom.
self.num_interactions_condoms += 1
use_condom = True

if self.people[a].is_infected or self.people[b].is_infected:
is_transmission = numpy.random.random() <= self.prob_transmit_condom
else:
is_transmission = False
else:
# Not using a condom.
use_condom = False
if self.people[a].is_infected or self.people[b].is_infected:
is_transmission = numpy.random.random() <= self.prob_transmit
else:
is_transmission = False

# Now infect.
self.history_interactions.append((self.t, a, b, use_condom, is_transmission))
if is_transmission:
self.people[a].is_infected = True
self.people[b].is_infected = True

def get_num_infected(self):
"""
Get the number of infected persons.
"""
# Count
infected = 0
for person in self.people:
if person.is_infected:
infected += 1

return infected

def step(self):
"""
Model step function.
"""

# "Interact" agents.
self.step_interact()

# Move agents
self.step_move()

# Increment steps and track history.
self.t += 1
self.history_space.append(copy.deepcopy(self.space))
self.history_space_infected.append(self.get_space_infected())
self.num_infected = self.get_num_infected()
self.history_num_infected.append(self.num_infected)
self.history_num_interactions.append(self.num_interactions)
self.history_num_interactions_condoms.append(self.num_interactions_condoms)

def get_space_infected(self, t=None):
"""
Return a projection of the space that shows which cells have an infected person.
"""
if t == None:
# Initialize empty
infected_space = numpy.zeros_like(self.space)

# Iterate over persons and set.
for p in self.people:
x, y = self.get_person_position(p.person_id)
if p.is_infected:
infected_space[x, y] = +1
else:
infected_space[x, y] = -1

# Return
return infected_space
else:
# Return historical step
return self.history_space_infected[t]


## Results¶

### Overview¶

We hope to present plots that demonstrate how sweeping over percentage changes in subsidy rates at: [0.8,0.9,1.0,1.1,1.1,1.2,1.3] will result in changes to HIV infection rates. These plots will include: 1) HIV Infection Rate by Condom Subsidy Rate at time step 100, 2) HIV Infection Rate over time by Condom Subsidy rate for 3 selected case

### Hypothetical Results¶

While adjusting condom subsidy up, we expect that condom use will increase and subsequently, the HIV infection rates will decrease over time. We believe that there may be a limit for which HIV infection rates will decrease to, and then remain at over time.