Application of PanDict System Based on EPSEIRV and SI3R Models in Epidemic Forecasting and Healthcare Resource Planning

Almost all mathematics-based disease transmission models are variants of the SIR model. This section explains the SIR model and presents some popular variations of it [4]. The SIR model separates the population into three compartments: Susceptible, Infectious, and Removed. Everyone who has neither obtained immunity against the disease nor been infected with the disease is considered Susceptible; everyone who is infected and can infect others is regarded to be Infectious, and everyone who is immune to the disease is defined in the Removed category.

The SEIRV model implemented two additional compartments to the SIR model, Exposed and vacant. The original Infectious group in the SIR model is elaborated into two, taking patients’ latent period into account; the Exposed group contains all who have been infected but are not yet infectious. At the same time, the Infectious group includes those who were infected and have become infectious. The Vaccinated compartment comprises of people who have been vaccinated but have not yet gained immunity (those who have gained immunity are considered Removed). A more complete review of the SEIRV model is provided in the Method section.

In short, the SEIRV model has an exponent on the infection expression, α, which has no physical meaning, so it can’t be determined with precision at all. It is pre-set to be around 1.2, a value that works best for most kinds of diseases. Thus, with α predetermined, the model only contains one degree of freedom: the contagiousness measure, β, which significantly undermines its accuracy. In contrast, the author improved the model by discarding the guessed variables and introducing two additional parameters derived from manipulating probability equations of encounters and the fraction of time people spend interacting with others. This improvement allowed us to accurately model the spread of the Omicron in January.

Most researchers use mathematical or biological approaches to predict the emergence of future variants of a disease. This section explores the deficiencies of each approach and how our system resolves this problem. The biological approach to predicting future variants requires an extensive amount of experimentation and often takes a large amount of time before attaining adequate information. In the case of Omicron, the infectious disease expert, Dr. David Aronoff, who attempted to use biological methods to predict future variant emergence, said after the surge of Omicron that they still “really don’t know if there’s going to be another variant that may create a lot of havoc.”8 This more mathematics-based article by Minus Van Baalen and Maurice Sabelis explored the competition between the resident strand and a mutant of a virus [11]. The study also explained the impact of that competition on the evolution of the virulence, as well as the infectiousness, of a disease. However, the study stopped on the theoretical level and did not provide a usable model for predicting variants. Thus, the author devised the SI3R model to simulate the competition between the resident strand and a hypothetical mutant of a virus to explore the conditions under which new variants could emerge.

Before our project, the Brown School of Public Health was the only group that attempted to calculate the projected demand for hospitalization resources [12].However, their study did not implement a disease transmission model that could estimate the spread of the disease; instead, they guessed the infected population and time and then calculated future demand according to the guesses. Thus, it has no specificity to the disease and will not be instructive to policymakers who need predictions of exactly where and when hospitals will hit their capacity to prepare for the exponential growth of the number of patients.

The system aims to help minimize the damage of a newly emerged epidemic. It consists of three essential modules, each addressing one of the previous studies’ problems.

As shown in Figure 1, the process of the system is as follows. The user first inputs the population statistics of the target country and enters the date of the first case in each local community. The basic parameters of the newly emerged disease are also required. Those mainly include the incubation period, infectious period, and vaccine efficacy. Subsequently, the system produces and displays an accurate prediction of the spread of the new virus according to those parameters. With the projected infection curve, the second module will test the condition and probability of the emergence of a new variant using the SI3R model. Lastly, the system uses the projected infection curve to calculate the demand for health resources in each individual county of the target country and displays the projected data on a map (you can download the actual numbers as well).

The SEIRV model is based on the following assumptions:

The total population remains constant (new births and deaths have trivial impacts and are not taken into account).

The exponent α in (1), (2), (5) has no physical meaning and thus cannot be obtained in any way other than estimating with infection data. For most kinds of diseases, the value of α = 1.2 works the best, so it was set to be around 1.2 for novel diseases as well without any other supporting experiment or evidence.

The model has only one degree of freedom (beta) with the exponent alpha predetermined; this lack of specificity significantly reduces its accuracy and robustness. In addition, this model does not account for changes in population density and social distancing. Thus, the author made the following changes to the model.

This section summarizes the derivation of the daily infection expression.

Assuming that the probability of one person meeting an infectious individual and capturing the disease is ^λ and that an average person spends μ fraction of their time interacting with others, then the probability of the person catching the disease at all equals to: 6 μ(1−(1−λ)I)

where I equal to the number of Infectious agents [13]. Because λ represents the probability of meeting one infectious individual and simultaneously receiving the disease, it should be almost infinitesimally small. Since e=(1+1n)n when n approaches infinity, author can approximate e=(1−λ)1λ. Then, author may rearrange the expression as follows: 7 μ(1−(1−λ)I)=μ(1−(1−λ)−1λ−λI)=μ(1−e−λI)

This change in the expression connects daily infection numbers with population density and exposed time. Exposed time is associated with the model by μ, while this paragraph illustrates the model’s connection to population density. Suppose r is the radius of a person’s daily activity range, p is the probability of catching the disease upon encounter, N is the population of the target community, and A is the area. Then, the author can express λ as pπr2A , Substituting A with ND . D represents the population density. Finally, author yield λ=pπr2DN . Showing that the infection coefficient λ is directly proportional to the population density.

Thus, the system of differential equations in the EPSEIRV model is as follows: 8 dSdt=−μ(1−e−pπr2DNI)S−ψεS 9 dEdt=μ(1−e−pπr2DNI)(S+V)−σE 10 dIdt=σE−γI 11 dRdt=γI+ıV 12 dVdt=ψεS−μ(1−e−pπr2DNI)V−ıV

S represents the number of people who are susceptible to the disease, E represents the number of people who have caught the disease but are not yet infectious, I represents the number of people who are infectious, R represents the number of people who have gained immunity against the disease, V represents the number of people who are under vaccination but are not fully vaccinated, t represents time in days, ψ represents the proportion of newly vaccinated individuals to the susceptible population, ε represents vaccine efficacy, σ represents the inverse of disease incubation period in days, γ represents the inverse of disease infectious period in days, ι represents the inverse of the time needed to gain immunity fro vaccines in days, p represents the probability of one person capturing the disease upon encounter with an infectious agent, D represents population density, N represents the number of people in population, r represents the radius of an average person’s daily activity range, and finally, μ represents the fraction of time an average person spends interacting with others. μ and p are social parameters in this model, while the rest are basic parameters.

The following chart elucidates the changing relationship between each compartment:

On day 1, E is set to 1, R is set to the number of people who have received a booster shot before day –14 (14 days prior to the first case), V is set to the number of people who received a booster shot between day –14 and day 0, I is set to 0, and S is set to N – E –V – R.

According to Centers for Disease Control and Prevention (CDC) investigations, the incubation period of Omicron is approximately three days, and the mean infectious period of Omicron is about 11 days [14]. CDC reports also reveal that the effectiveness of unboosted vaccines is insufficiently low, and the British Broadcasting Corporation (BBC) claims that the efficiency of booster shots is roughly 80% [15][16]; therefore, the author decided only to include booster shots as an effective type of vaccination. In the United States, the boosted population has an average daily increase of 600,000 (0.18% of the entire population), while in South Africa, no one has received a booster shot yet, so the author decided to abandon the Vaccinated category when modeling the spread in South Africa [3].

As shown by the United States Department of Labor statistics, below is a list of major daily activities and time spent by an average American (in hours) [17].

The activities marked red are the ones that potentially involve in-person interactions with others. However, for each activity, only a portion of the times listed above will imply in-person interactions. Supporting facts are as follows:

As shown in TABLE I. , Roughly 80 percent of the population shops online [18]. Forty-two percent of the workforce works from home [17]. About 40 percent of socializing time is online [19]. Private cars dominate the American Commute. Only 5% of US commuters use public transportation [20]. Forty-six percent of students receive only online instruction [21].

Therefore, the number of hours that involve potential interactions with others can be calculated as: 0.38×0.2+ 0.14 + 3.02 × 0.58 + 0.18 × 0.54 + 0.18 + 0.54 × 0.6 + 0.37 + 0.79 × 0.05 ≈ 2.97 hours, which means μ=2.9724 With all the parameters 24 specified, author graphed this system of differential equations by compiling it into a Python program. Then, the author utilized the least rooted mean squared error (RMSE) to determine the best representative p-value and the most accurate model for the spread of Omicron in the US. The amount of time people spends interacting with others in South Africa is not much different from that of the United States because it compares two groups with large populations and similar pandemic-control regulations.

Upon the emergence of a novel virus, not only are authors concerned about the spread of the resident strand of the virus, but the public is also perturbed by the possibility of the appearance of new variants, particularly when the author can’t obtain sufficient information about when and under what conditions new variants may emerge. Since little to no work has been done to investigate the competition between a mutant and a resident, the author devised the SI3R model to simulate the competition and thus determine the conditions under which a new variant might emerge.

The model consists of five compartments: Susceptible, Infectious1, Infectious2, Infectious, and Removed. The model’s assumptions include those of the improved SEIRV model and that the mutated strand cannot evade the public’s immunity against the resident variant. Therefore, the Susceptible compartment represents the portion of the population that has caught neither of the strands and is thus prone to receive the virus. Infectious contains those who have been infected with the resident strand and are capable of spreading it. Individuals in the Infectious2 compartment can spread the mutant, while those in the Infectious compartment can transmit both strands. Since both strands, the resident and the mutant, according to the prediction, cannot escape the immunity against the other strand, individuals removed from any of the three infectious compartments would be defined in Removed. This system of equations is not extremely detailed but is sufficiently functional to generate general results. More future work could be implemented to enhance the accuracy, for example, by adding the Exposed categories.

As established above, the expression for daily infection number of one variant can be modeled by μ(1−e−αI)S . Thus, combining the infected number of the five possible transmission routes results in μ(5−e−a1I1−e−α1Ia−e−a2I2−e−α2Ia−e−αaIa)S . Then, dividing up the infected population into one of the three Infectious groups yields the correct model.

Below is the system of differential equations that describe the change of each compartment. 13 dSdt=−μ(5−e−a1I1−e−α1Ia−e−a2I2−e−α2Ia−e−αaIa)S 14 dI1dt=μ(2−e−a1I1−e−α1Ia)S−μ(1−e−a2I2)I1−θ1I1 15 dI2dt=μ(2−e−a2I2−e−α2Ia)S−μ(1−e−a1I1)I2−θ2I2 16 dIadt=μ(1−e−αaIa)s+μ(1−e−a2I2)I1+μ(1−e−a1I1)I2−θaIa 17 dRdt=θ1I1+θ2I2+θaIa

S represents the number of people who are susceptible to both the resident and the mutant, I1 represents the number of people who are infected by the resident strand, I₂ represents the number of people who are infected by the mutated strand, I_a represents the number of people who are infected by both strands, R represents the number of people who have gained immunity, θ_I represents the inverse of the infectious period of the resident strand, θ₂ represents the inverse of the infectious period of the mutated strand, θ_a represents the inverse of the infectious period of individuals carrying both strands, α₁ represents the resident strand’s transmission rate from a person in the Infectiousa compartment, α₂ represents the mutated strand’s transmission rate from a person in the Infectiousa compartment, α_a represents the probability of an individual who carries both strands transmits all two to a susceptible individual, a₁ represents the resident strand’s transmission rate from a person in the Infectious1 compartment. a₂ represents the mutated strand’s transmission rate from a person in the Infectious [2] compartment, and finally, μ represents the fraction of time an average person spends interacting with others. On day 1, I₁ is set to the number of patients in reality, I₂ is set to 1, R is set to the number of people who have been infected and removed in reality, I_a is set to 0, and S is set to (Population) –I₁ –I₂ –I_a –R.

The best-fitting p for the data obtained in the United States equals to 3.57×10-8. Please note that the author used positive test percentages instead of new daily cases because the percentage better represents the status of the entire population, for only a small portion of the population is tested each day. In addition, the author subtracted the percentage of Delta variant positive cases (used the model on Delta as well) from the total positive percentage. Also, the author assumed day 0 to be November 22, 2021, since the first detected case in the US is a traveler who returned from South Africa on November 22 [22].

As shown in Figure 5, in which the vertical and horizontal axes respectively represent infected population percentage and time, the red line is the data the author used to generate our prediction (green line), while the purple line represents the real-life data of the ensuing days. Our prediction was generated on January 6 and based on the data before that day. Our model accurately predicted the peak of the infected percentage and produced a projected infection curve that is almost identical to that in real life. The dotted lines mark the error of our predicted trajectory, which, shown in the picture, is negligibly small.

Figure 6 is the yielded real time reproduction number of Omicron in the US as a function of time by the EPSEIRV model. This concludes that the R0 value of Omicron in the US is approximately 18.8, and its real-time reproduction number (R(t)) decreases as a larger portion of the population becomes infected or removed. This is a reliable R0 value because the infection curve of this R0 value perfectly models the actual infection data. (Note: R0 is a concept in epidemiology that estimates an infectious agent’s propensity for epidemic transmission. Simply explained, the R0 value of a disease means how many secondary infections were caused by the very first case in a fully susceptible population, while R(t) is defined as the average secondary infections caused by one patient relative to time [23].) An R0 value of 18.8 is extremely high for a disease, which is consistent with its absurdly quick circulation speed in the US. Before the author published our data on Medium.com, the CDC announced that the R0 value of Omicron was about 75. Now, with more projects examining the R0 number of Omicron in the US, an R0 number above 13 for Omicron has become somewhat of a consensus, reaffirming our produced result.

In contrast, Figure 7 shows the performance of the original SEIRV model when α, according to its published essay, is set to 1.2. All other parameters were set the same way as the EPSEIRV model. While also using the same amount of data, the SEIRV model performs significantly worse than our proposed EPSEIRV model (the shaded region also represents its prediction error). Apparently, Omicron’s absurdly high contagiousness is not well considered in the 1.2 value of alpha in the original model.

Figure 6 models the increase of accumulative positive Omicron cases in the United States; it is found that the increase of accumulative positive Omicron cases will be slow until late December 2021. However, it will increase rapidly by March 2022, which more directly shows that most of the population will be infected with Omicron before the end of March 2022. After that, the cumulative number of cases will not increase and level off.

Since the existence of a mutant is hypothetical, the author can’t determine the specific parameters of it. However, the model still produced illustrative results. Assuming that on March 5, 2022, a new mutant of Sars-CoV-2 appears, thus setting day 1 to March 5, 2022, the author obtained Figure 7. The purple line represents the Removed category, the orange line is Infectious1, the minuscule green tip is Infectious2, and the almost straight red line is Infectiousa. This study tested with a range of input parameters, from absurdly high transmission rate to low transmission rate and from long infectious periods to short infectious periods. Nonetheless, it had a negligible impact on the growth of the number of patients in each Infectious compartment, for both strands died out very quickly in all scenarios due to the large number of removed individuals. Thus, under the assumptions of the model, it is not likely that a new mutant of Omicron can prevail. In conclusion, assuming that no drastic changes happen to the US population, for a new variant to prevail, it has to overcome the immunity against Omicron.

For demonstration and testing purposes, the author ran the system based on Omicron data prior to January 6, 2022. Figure 8 shows the need for hospitalization resources in each state between 50-60 days since the first case. Users of the system can input different day numbers to access the estimation for that day with a slider. Again, the author only finished the Northeast region of the United States, for it is time-consuming to enter the parameters mentioned in Methods section. Future work may expand the system to the US or even around the world.