As the primary function of the model was to determine the number of children in each state of health at a given point in time, the model’s core consisted of a basic Markov model. This category of model has been used for various types of probabilistic modeling over the past 20 years, including epidemiology modeling, especially when the model has a defined number of outcomes, or states.15 The Markov model used for the creation of the coverage map for Yemen was composed of four different states (Healthy, Sick, In-treatment, and Deceased) for children to move between based on a set of probabilistic transition rates. The model rests on the following assumptions: any sick person that decides to seek treatment will contact a health care provider in community, in a mobile team or at a health facility, and will be treated; to reach the In-treatment state one must have previously resided in the Sick state; the Deceased state is an absorbing state. The health facility data collected through a third party monitoring exercise provided insight on the number of children under five years old seen and treated for diarrheal diseases, the treatment mortality rate of Zn/ORS solutions used in the region, the efficacy of Zn/ORS solutions in treating the disease and the percentage of the population the facilities monitored are estimated to have served. This information was filtered to examine only diarrheal diseases and to eliminate any incomplete reports where the total number of reported children treated for diarrheal diseases did not equal the sum of all the individual children reported to have diarrheal diseases. Coupled with the most recent overall mortality rates in the literature from the WHO and an estimated incidence rate for Yemen found in the literature, the transitions rates were created to allow the model’s output to match the previously mentioned data.11,16 The final transition probabilities were as listed in Table 1. An overview of the terms used, as well as their values and meaning are provided below in Table 2. These are described in more detail later in the manuscript.
Table1: Transition Rate Probability Table
State Transition
|
Transition Probability
|
pHH Healthy to Healthy
|
.7601
|
pHS Healthy to Sick
|
.2388
|
pHT Healthy to Treated
|
0
|
pHD Healthy to Deceased
|
.0011
|
pSH Sick to Healthy
|
.6968
|
pSS Sick to Sick
|
(1-(YT+6.2712)/9)
|
pST Sick to Treated
|
(YT-.01296834)/9
|
pSD Sick to Deceased
|
.00092631
|
pTH Treated to Healthy
|
.9902
|
pTS Treated to Sick
|
.00961
|
pTT Treated to Treated
|
0
|
pTD Treated to Deceased
|
.00019
|
pDH Deceased to Healthy
|
0
|
pDS Deceased to Sick
|
0
|
pDT Deceased to Treated
|
0
|
pDD Deceased to Deceased
|
1
|
List of the probabilities of a child transitioning from one state of health to another in the model.
Table 2: Table of Terms
Object Notation
|
Summary
|
Value
|
Susceptible Population
|
Total under 5 child population
|
358498
|
Cycles
|
Number of weeks the simulation runs for
|
52
|
Maxroad
|
Max value of roads
|
variable
|
Roadmin
|
Minimum value or roads due to washout
|
equation
|
Bridge
|
Percent of open road segments with open bridges
|
variable
|
Nonbridgemin
|
Minimum road value with operating bridges
|
equation
|
Bridgecuttoff
|
Weather threshold where lower quality roads may begin to experience washout
|
0.5
|
Scalemax
|
Parameter determining how long infrastructure stays at max value
|
variable
|
Weatherweight
|
Weighting of weather on transition rate
|
0.3
|
Roadweight
|
Weighting of infrastructure on transition rate
|
0.7
|
Y(t)
|
Weather function
|
Equation
|
Road(t)
|
Road and bridge infrastructure function
|
Equation
|
YT(t)
|
Transition probability equation from Sick to Treated
|
Equation
|
Overview of the variables and parameters used throughout the model.
To help account for non-cholera based diarrheal diseases the transition rate from healthy to sick was further subdivided between general sources of the diseases and Rotavirus, the most common source which is noted as the cause of between 35-60 % of enteric diseases in children under five years old.18 Rotavirus infections confer natural immunity, protecting against 87% of severe diarrhea cases which increases with subsequent infections.18 Thus, each rotavirus infection after the first is estimated to result in a 0.13n chance that the child would become sick from the virus alone where n is the number of previous infections a child has contracted.
Along with this information, other types of data outside of traditional health data were incorporated into the model’s transition rates, most notably for the probability determining the likelihood that someone will be willing and able to seek out treatment and thus move from the Sick state to the In-treatment state. As such, this influences the probability that a sick person will remain sick. The state of the infrastructure and the estimated weather conditions for that time period allows for the creation of an equation incorporating the time varying nature of these data types into the probabilistic decision process of the model. Thus, the model is not only probabilistic but also time varying.
The state of the weather and the quality of the roads and bridges in Yemen, which are already affected by and dependent on the state of the conflict and so serve as a proxy for the severity of the conflict, have been shown to be related functions. The weather component is assumed to be a sinusoidal function with a period of one year. This weather function was created as a normalized function of best fit from monthly precipitation data for Yemen from 1901 to 2016 published by the World Bank.19 The range of probabilities vary between 0, which in the context of the model would indicate a perfect storm during the heaviest rainfall of the year preventing anyone from traveling, to 1, which would result in anyone being able to access roads leading to health services during the driest week of the year. This function is further multiplied by a set constant of 97.84% which is to account for the fact that there are 21.6 airstrikes for every 1000 people according to data published by the Yemen Data Project Organization.20 Like the rains, this can deter people from traveling and further destroy infrastructure, albeit much more directly.
It is estimated that Yemen has about 50,000 km of roads.21 As the country does not have a functioning rail system to aid in transportation, these roads are the primary means of transportation for people within the country.21,22 Despite the importance of these roads, reports from the World Bank estimated that only 28% were all-weather paved before the conflict’s start, with only 11% of rural roads being paved.21 These nonpaved road segments have been reported by USAID to be damaged both by airstrikes and seasonal rains which has led to flooding, hindering travel during the rainy season.23 The model was constructed on the assumption that the rains predominantly affect the coastal regions to the south and west of the country and so only 50% of the unpaved roads potentially experience wash out from the worst of the rains. The bridges have also been affected by both the weather and the war with air strikes and fighting destroying bridges while many others have their access restricted according to maps published by the World Food Program and the Logistics Cluster.24 These restricted bridges were noted to have detours that “may not be accessible during the rainy season”.24 Thus, when the rains begin the weather function will start to decrease. When the function drops below “bridgecuttoff”, a tunable threshold, a percentage of the roads corresponding to passable bridge detours that were previously opened will close. This percentage was designed to scale with the weather and vary from its max and minimum values due to the conflict.
A road can have multiple different conditions depending on which section of the road was examined. Because of this, the roads shown on the maps published by the Logistic Cluster and World Food Program were subdivided into individual road segments. A road segment was defined as a unique stretch of road that connects between marked communities or another road segment. As the road function was dependent on the weather it too was cyclical. While its maximum value occurred when there were no weather based restrictions on travel, this value ultimately was determined from the extent of the conflict as described by Equation 1. These values were based on how extensive the conflict was during the period examined while the weather provided more regular variation determining how useable these open roads were.
Roadmax=1-(% closed road due to conflict)-(% road with impassable bridge) (1)
The minimum was calculated as the percent of road segments left functional after the assumed max wash-out minus the percent of road segments that have a weather dependent detour as shown in Equation 2 below.
Roadmin=.64×Roadmax-bridge (2)
With these extrema determined, the road function is as noted below in Equation 3.
Road〖(t)=〗 〖min{〗〖([Y〖(t) ×(scalemax-nonbridgemin)+nonbridgemin]-bridgedown〖(t)),〗 〗 MaxRoad}〗 (3)
Bridgedown is defined as Equation 4 below.
The overall transition probability function was then created as a weighted average of the weather function, Y, and the infrastructure and conflict function, Road, as seen in Equation 5.
With the weather and road components combined to form the transition rate function for the probability that a person will travel to seek treatment and thus change from the Sick state to the Treated state, it became possible to monitor the percent coverage. It was assumed that each person that made it to a treatment center (Sick to In-treatment transition states) was treated with Zn /ORS while medical supplies were available. As this transition occurred only after a person became sick, the percent treatment coverage was calculated as:
Since those infected during week t would only be transitioning into the In-treatment state during the following week, the numerator’s week number must be one ahead of the denominator for the calculation.