Patient data from this study was collected from various reports in literature on the frequencies of symptoms in COVID-19, influenza, MERS, and SARS (Supplemental Tables 1, 2). Each dataset was used either to approximate order of symptoms, to confirm our results, or to analyze first symptoms in COVID-19 or influenza. For all of these applications, we used the reported patient data to simulate patients with various combinations of symptoms experienced and then applied the simulated data to perform the analyses.

The main dataset of COVID-19 patients of the World Health Organization, containing 55,924 confirmed cases, was obtained through review of national and local governmental reports and observations made during visits to areas with infected individuals in China that occurred from February 16 to 24, 2020 (8). A confirmation dataset of COVID-19 patients, containing 1,099 confirmed cases, was obtained by the China Medical Treatment Expert Group for COVID-19 from medical records and other compiled data of hospitalized patients and outpatients that were diagnosed with COVID-19. This data was reported to the National Health Commission of China from December 11, 2019 to January 29, 2020 (16). For both COVID-19 datasets, myalgia was reported as myalgia or arthralgia. We assumed that most patients with myalgia also had arthralgia, and therefore we used the frequency of myalgia or arthralgia as a frequency for myalgia when simulating data. The influenza dataset, containing 2,470 confirmed cases, was collected by researchers at the University of Michigan from a retrospective pooled analysis of mostly unvaccinated patients participating in phase 2 and 3 clinical trials that were conducted in North America, Europe, and the Southern Hemisphere from 1994 to 1998 (6). This group of patients has a mean age of 35 and each exhibited multiple symptoms. Vomiting and diarrhea were not reported in this influenza dataset, but they are common among respiratory disease. Although adult patients at times may experience vomiting and diarrhea when infected with influenza, these symptoms are rare (17). Therefore, we approximate the frequency of these symptoms as 0.010 in this case. The datasets representing symptom frequency in MERS, containing 245 patients, and SARS, containing 357 patients, were collected on admission and were reported as clinical data from physicians, Dr. Yin, at the Beijing Chao-Yang Hospital and Dr. Wunderink, at the Northwestern University Feinberg School of Medicine (7). The patients included in these datasets varied in age and pre-existing conditions. In the cases of SARS, the patients tended to be younger and have fewer pre-existing conditions than in the cases of MERS.

We used initial frequency data of MERS and SARS to further ascertain early symptoms of disease. The MERS initial symptom frequency dataset, containing 45 confirmed cases, was collected from electronic medical records at the Samsung Medical Center in Seoul, South Korea that contained onset symptom data about patients in the 2015 Korean MERS outbreak (Supplemental Table 3) (18). The SARS initial symptom frequency dataset, containing 144 confirmed cases, was collected from hospital records including information of early symptoms in patients dating from March 7 to April 10, 2003 during an outbreak in the greater Toronto area (Supplemental Table 4) (19).

Lastly, two additional datasets were collected to determine the utility of using first symptoms as early indicators of COVID-19 and influenza. The COVID-19 dataset used, containing 138 patients, was independent of all prior COVID-19 datasets. This data was obtained from electronic medical records of patients admitted to the Zhongnan Hospital of Wuhan University from January 1 to 28, 2020 (20). The symptom data was collected at onset of disease and all patients experienced pneumonia due to COVID-19. In this dataset, nausea and vomiting were reported separately for COVID-19. We assumed that most patients who experience vomiting, which is reported with a frequency of 0.036, also experience nausea, which is reported with a frequency of 0.101, and therefore to simulate the data, we defined the frequency for nausea/vomiting as 0.101. The influenza dataset used reported 20 confirmed cases of influenza and 400 confirmed negative cases of influenza and is independent from any other influenza dataset we used (21). The symptom data was collected through questionnaires and observations by medical professionals during the influenza seasons of 2006 and 2007 of infected patients admitted at the Department of Internal Medicine and Infectious Diseases and the Department of Pulmonology at the University Medical Center Utrecht. Like the other influenza dataset described above, vomiting and diarrhea were not reported in this dataset. So, we once again assumed the frequency of these symptoms to be 0.010 (17). Because this study was conducted in 2006 and 2007, prior to the COVID-19 outbreak, we assumed these patients were negative for COVID-19 as well. So, this 400-patient group was used as the dataset that represents individuals negative for both COVID-19 and influenza (Supplemental Table 5).

The Stochastic Progression Model was built in R under version 3.5.2 and was illustrated by using the hasse function in the hasseDiagram_0.1.3 library (code available online: https://github.com/j-larsen/Stochastic_Progression_of_COVID-19_Symptoms) (22, 23). Each respiratory disease report was represented by a corresponding data frame, with columns as symptoms, one row as the frequency of the symptoms observed in the study, and the other row as the frequency multiplied by 1,000. The multiple of the frequency is defined as the frequency count, which represents the probability of a symptom in a theoretical sample size of 1,000 simulated patients. Additionally, the state of an individual is displayed through a character array of ones and zeros, where ones represent the presence of a symptom and zeroes represent its absence. This process of simulating a symptom is analogous to a jar of marbles of either two colors. The probability of pulling one color of marble (i.e., a specific symptom) is illustrated by the frequency count because the total number of marbles in the jar is 1,000 and the frequency count for each is the number of the specific color of marbles in the jar.

We then simulated data of 500,000 patients, by randomly selecting if a patient has or does not have a symptom using the procedure described above and storing that information in a data frame that represents patients as rows and symptoms as columns. We assumed the occurrence of symptoms are random and independent. Considering these assumptions, we built the character arrays by applying the jar of marbles method for each simulated patient. The method repeats for each patient and involves pulling a marble from a series of jars representing each symptom. The information from each randomly pulled marble is stored in the corresponding cell of the character array in the correct column representing the symptom and the row representing the simulated patient. This process is repeated for all 500,000 simulated patients for all symptoms.

The Stochastic Progression Model is illustrated as a directed acyclic graph with nodes, representing the power set of Boolean vectors. The power sets of Boolean vectors each represent a possible state of a patient by noting the absence or presence of specific symptoms. The edges, which illustrate the transition from one state to another, were selected specifically using key definitions and assumptions to create a poset. We defined the states at the nodes as symptoms that a patient has experienced up until this point. We created and directed edges from states with fewer symptoms to more starting at the minimum set of a Boolean vector of all zeros, which indicates a person with no symptoms. First, we assume that each symptom occurs one at a time, even if the difference in time is infinitesimal. With this assumption, a node can only be directed to other nodes that denote the same set of symptoms plus one additional symptom. Second, we assume that if a patient does not digress and does not die, they will eventually acquire all symptoms reaching the maximum set of a Boolean vector, which represents a patient that has exhibited all symptoms. Applying these assumptions to form the directed acyclic graph creates a Hasse Diagram of a graded poset that follows a Markov Process altogether comprising the Stochastic Progression Model.

The nodes in the Hasse Diagram represent states of a patient by indicating the specific symptoms exhibited, and the edges represent transitions between these states. Therefore, we next needed to apply state probabilities to each node and transition probabilities to the directed edges. First, we labeled each simulated patient by summing the respective Boolean vector to find the number of symptoms for each patient. Then, to get the state probability of each node, we divided the number of simulated patients that are represented by the current Boolean vector by the total number of patients who have the same number of symptoms. To approximate the transition probability between two nodes (originating and terminating), we divided the number of simulated patients that are represented by the terminating node by the number of simulated patients that are represented by nodes characterized by the same number of symptoms as the terminating node, including the terminating node. The error of each node is determined by the sum of the products of the transition probabilities leading to that node subtracted from the state probability of the node. Then, the error of each implementation of the model was defined as the error of the node with the highest absolute value of error (Supplemental Figures 2–13). The transition probabilities signify the likelihoods of transitions from one node to another, and the aggregates of the transition probabilities in a sequence represent the likelihoods of the paths. These paths illustrate the order of symptoms when infected with a respiratory disease by observing the stepwise addition of symptoms when traversing down nodes in the path. The most and least likely paths were determined using a greedy algorithmic approach. This approach consists of selecting local maximum or minimum edges stepwise, which results in a most and least likely path, respectively. If the maximum (or minimum) transition probability from a specific node was within error of other transition probabilities of edges from the same originating node, we grouped the terminating nodes when finding the most (or least) likely path. In these cases, we could not distinguish a difference in likelihood between these specific transitions. The paths create a possible order of symptoms via the poset, each having a specific likelihood of occurrence.

The WHO-China Joint Report from February 16 to 24, 2020 includes rates of symptom occurrence at presentation from 55,924 confirmed cases of COVID-19 (8). We identified symptoms that were easily discernible or objective (i.e., fever, cough, diarrhea, and nausea/vomiting) in comparison to other reported symptoms, such as inflammations of blood vessel epithelia (24), neurological effects (25), and rash-like symptoms (26). These symptoms are also common in other respiratory diseases. Thus, we chose to implement these four symptoms in the Stochastic Progression Model (Supplemental Table 1). To confirm the validity of the model, we first determined the possible sequences of symptom occurrence when the probabilities are uniformly random for each symptom. In addition to all possible orders of occurrence of the four symptoms, the diagram displays the most and least likely paths of the four symptoms, depicted by red lines and blue lines, respectively (Figures 1A,B). The most and least likely paths describe the most and least likely series of symptoms that a random infected person from the population in the dataset may experience. In this case, each possible path is equally likely, with no path having any higher probability than any other.

We then created another implementation of the Stochastic Progression Model and utilized the data in the WHO-China Joint Report (COVID-19 with N = 55,924) (8). With this implementation, we determined the most and least likely paths (Figure 1C). In this case, a person infected with COVID-19 is most likely to experience symptoms in the order of fever, cough, nausea/vomiting, then diarrhea (Figure 1D). The least likely path starts at diarrhea and nausea/vomiting and is followed by cough, and finally fever (Figure 1E). We confirmed these results with a smaller dataset (COVID-19 with N = 1,099) (Figures 1D,E, and Supplemental Figure 1) (16). The likelihoods of transitioning to fever, 0.769, and then to cough, 0.958, are high, and these observations indicate that a large portion of infected symptomatic patients may follow this path. Finally, this implementation of the model predicts that nausea/vomiting occurs before diarrhea. These two results suggest that in patients with SARS-CoV-2, the body first develops fever, then upper respiratory symptoms and finally symptoms of the upper then lower gastrointestinal (GI) tract.

To further investigate these symptom paths, we implemented the Stochastic Progression Model with the main dataset (COVID-19 with N = 55,924) (8), to determine the likely downstream paths when the first one or two symptoms are forced to a certain state (Figures 1F–I). The gray lines represent the “forced” paths. The rest of the paths were determined as before with a greedy algorithmic approach. We found that the most likely orders of the downstream path are consistent with the most likely orders of the unforced paths. Even if the first symptom is forced to be an unlikely one (e.g., diarrhea), the downstream paths maintain the most likely order of the other three symptoms that we originally determined (Figure 1F). Similarly, the GI tract effects occur first in the forced least likely paths (Figure 1G). When forcing the path one step further by predetermining the first two symptoms for both the most and least likely paths, the findings remain the same (Figures 1H,I).

The confirmation dataset of COVID-19 cases (N = 1,099) separates the reported 1,099 cases between severe and non-severe patients as designated on admission (16). To investigate the effects of severity on the order of discernible symptoms, we implemented each set of cases separately using the Stochastic Progression Model. We found that the most and least likely paths are identical in severe and non-severe cases and to our original findings above (Figure 2). To illustrate the similarities, the largest difference in likelihood is observed when transitioning from no symptoms to fever in the most likely path. In severe and non-severe cases, the probability is 0.775 and 0.818, respectively, indicating a difference of 0.043. These results suggest that severity does not affect the order of discernible symptoms, and they are consistent with the hypothesis of fever as the first symptom of COVID-19.

The four discernible symptoms are objective and relatively easy for patients and clinicians to confirm. So, we developed implementations of the Stochastic Progression Model using these symptoms to determine the most likely and least likely paths for four respiratory diseases: COVID-19, influenza, MERS, and SARS (Figures 3A–D) (6–8). The most likely order of occurrence of symptoms in COVID-19 is fever, cough, nausea/vomiting, and diarrhea (Figure 3A). This path is identical to influenza except the order of the initial two symptoms is switched (Figure 3B). On the other hand, the predicted most likely paths (i.e., fever, cough, diarrhea, and then nausea/vomiting) are the same for MERS and SARS (Figures 3C,D). This order has one difference from the most likely path in COVID-19 in that the order of the final two symptoms are reversed. The least likely path of MERS starts with either nausea/vomiting or diarrhea as the first step. These steps are followed by cough, and finally fever. In contrast, the least likely path of SARS is cough, nausea/vomiting, and diarrhea in any order, and then finally fever. However, the least likely path of symptoms in COVID-19 is the same as the least likely path in MERS, and the least likely path of influenza is unique compared to the other diseases. It is not detectable whether nausea/vomiting or diarrhea are the first symptoms in influenza, but after these two, the least likely path continues from there to fever then cough. This observation further illustrates the strong link of cough to influenza. As for coronavirus-related diseases, the strongest first indicator is fever followed by cough.

Although active surveillance of the order discernible symptoms (i.e., fever, cough, nausea/vomiting, and diarrhea) could be useful due to the distinctive most and least likely paths that we determined, we expanded our analysis to the seven symptoms commonly observed in all four respiratory diseases studied here. So, we created a second set of symptoms that amends sore throat, myalgia, and headache to the original set of symptoms (Supplemental Table 2). The three additional symptoms are more subjective (6–8). The seven-symptom implementation of the Stochastic Progression Model of COVID-19 shows that these additional symptoms did not perturb our initial ordering of fever, coughing, nausea/vomiting, and diarrhea, but instead added another level of intricacy in the middle of the likely paths (Figure 4). We still find that the most likely path first transitions to fever, indicating that fever is the most likely first symptom. From there, the most likely next symptom is cough once again. Then, we observe an undetectable difference in likelihood of transitioning to either sore throat, headache, or myalgia, indicating that all three are likely to occur next before proceeding. The final two nodes are consistent with the four-symptom order by indicating that nausea/vomiting then diarrhea occur last. Although this implementation is more complex because it has seven symptoms, it is consistent with our earlier findings. The most likely path of COVID-19 symptoms is fever, then cough, and next either sore throat, myalgia, or headache, followed by nausea/vomiting, and finally diarrhea, and this order is the same as the one indicated by the implementation developed from the confirmation dataset (COVID-19 with N = 1,099) (Figure 4) (16).

We also implemented the Stochastic Progression Model with the same seven symptoms in influenza, SARS, and MERS datasets to compare and contrast disease progression with that in COVID-19 (Figure 5) (6–8). The results for influenza indicate that cough or myalgia may occur first (Figure 5A). After these two symptoms occur, the order of symptoms is headache, sore throat and fever. Finally, vomiting/nausea and diarrhea have an undetectable difference in probability of occurring last. The MERS implementation displays a most likely path in which fever will occur first, followed by cough, headache, and then myalgia (Figure 5B). These are followed by an undetectable difference in likelihood of headache and diarrhea occurring. Finally, sore throat and nausea/vomiting will occur last with an undetectable difference. The implementation for SARS shows that fever is most likely to occur first, followed by an undetectable difference in transition probability of cough and myalgia, which is similar to the other coronavirus-related diseases (Figure 5C). Next, headache is most likely. Finally, diarrhea, sore throat and nausea/vomiting occur with an undetectable difference in likelihood.

To illustrate the uniqueness of the most likely path of COVID-19, we found the transition probabilities of the same path in the other respiratory diseases (Figure 6). When comparing and contrasting the probabilities, we found that the implementation representing COVID-19 strongly asserts that the first symptom will be fever and cough will soon follow because the transition probabilities are 0.731 and 0.783, respectively (Figure 6A), whereas the influenza implementation indicates that fever is very unlikely to occur first with a probability of only 0.035 (Figure 6B). Additionally, the implementations of MERS and SARS data also have a high likelihood of transitioning to fever first, with a probability of 0.627 and 0.988, respectively (Figures 6C,D). The second symptom of the most likely path of COVID-19 is cough, with a probability of 0.783, but the others do not have a similar high probability. For example, the respiratory disease with the highest probability at that transition is MERS at 0.536. However, after fever and cough, COVID-19 and the other three respiratory diseases have a similarly high likelihood of the three subjective symptoms (i.e., sore throat, headache, and myalgia). Finally, the most likely path of COVID-19 ends with nausea/vomiting and then diarrhea. These observations are consistent with the symptoms described by the CDC and support the notion that fever followed by cough seems highly likely to be diagnosed as COVID-19 (5).

Also, comparing the transition probabilities of paths in the same disease illustrates the significance of the most likely pathways. For example, the lowest transition probability in the most likely path of influenza is 0.578 (Figure 5A), whereas when analyzing influenza as it traverses down the most likely path of COVID-19, the transition probabilities observed are 0.5 or less (Figure 6B). However, in that same path, the transition probability from fever and cough to fever, cough, sore throat, headache, and myalgia is >0.999. This value displays how unlikely nausea/vomiting and diarrhea are to be initial symptoms of influenza. Additionally, when observing the most likely path of COVID-19, the first two symptoms seem to have a strong probability of occurring in the order of fever and then cough, with a likelihood of 0.731 (Figure 5A). However, the likelihood of cough occurring first in COVID-19 is 0.229, which is a low probability (Figure 5A). This observation further supports the hypothesis of fever occurring first and cough occurring second.

The COVID-19 and influenza implementations of the Stochastic Progression Model suggest that there is a high likelihood of fever and cough occurring first, respectively. We desired to find metrics quantifying the possible link between first symptom and these two diseases. So, we determined the recall and the selectivity when using the initial symptom as an indicator of COVID-19 or influenza, with all other possible diseases excluded in a theoretical patient population. First, we simulated patient datasets using reported data that were independent from all previous work that we integrated in our analyses above (Supplemental Table 5) (20). Two simulated patient datasets were created to analyze COVID-19 and influenza separately to portray the specific link of each disease with the corresponding initial symptom that we determined, fever and cough, respectively. The simulated data contained information about the patients' state of disease (COVID-19, influenza or not) and their first symptom experienced. Based on the information of the first symptom alone, we categorized the simulated patient data as infected with COVID-19 or not and influenza or not. The recall was calculated as the number of simulated patients that we correctly identified as having the disease over the number of simulated patients that truly had the disease (27). Selectivity was defined here as the number of simulated patients that we correctly identified as not having the disease over the number of simulated patients that truly did not have the disease (28). For both diseases, we performed this analysis for five simulated samples of different sizes, each containing 5% infected individuals. We repeated this process 10 times and calculated the average and standard deviation across each sample size for both COVID-19 and influenza (Tables 1, 2).

The recall ranges from 0.966 to 0.990 with a standard deviation of 0.031 and 0.021, respectively when analyzing the link between COVID-19 and fever as a first symptom. The maximum standard deviation of any sample size is 0.063 for the mean of 0.980. On the other hand, the selectivity of fever as a first symptom of COVID-19 ranges from 0.661 to 0.668 with a standard deviation of 0.030 and 0.020, respectively, and 0.030 is the maximum standard deviation with corresponding means of 0.661 and 0.665 (Table 1). As for cough as a first symptom of influenza, the recall ranges from 0.765 to 0.820 with corresponding standard deviations 0.092 and 0.067. The highest standard deviation is 0.110 with a mean of 0.810, and the selectivity ranges from 0.362 to 0.369 with standard deviations of 0.014 and 0.031, respectively, and the maximum standard deviation is 0.031 (Table 2).

The recall in both cases is lower than the selectivity, and this observation indicates that this analysis categorizes patients as infected when they are not, but the high recall indicates that most infected patients did align with the first symptom that we predicted. In the future, we expect to confirm this analysis with data on first symptoms, as opposed to simulated data, but the purpose of this analysis was to display that further study of order of symptoms might lead to earlier recognition.