A generalized differential equation compartmental model of infectious disease transmission

Abstract

For decades, mathematical models of disease transmission have provided researchers and public health officials with critical insights into the progression, control, and prevention of disease spread. Of these models, one of the most fundamental is the SIR differential equation model. However, this ubiquitous model has one significant and rarely acknowledged shortcoming: it is unable to account for a disease's true infectious period distribution. As the misspecification of such a biological characteristic is known to significantly affect model behavior, there is a need to develop new modeling approaches that capture such information. Therefore, we illustrate an innovative take on compartmental models, derived from their general formulation as systems of nonlinear Volterra integral equations, to capture a broader range of infectious period distributions, yet maintain the desirable formulation as systems of differential equations. Our work illustrates a compartmental model that captures any Erlang distributed duration of infection with only 3 differential equations, instead of the typical inflated model sizes required by traditional differential equation compartmental models, and a compartmental model that captures any mean, standard deviation, skewness, and kurtosis of an infectious period distribution with 4 differential equations. The significance of our work is that it opens up a new class of easy-to-use compartmental models to predict disease outbreaks that do not require a complete overhaul of existing theory, and thus provides a starting point for multiple research avenues of investigation under the contexts of mathematics, public health, and evolutionary biology.

A generalized ODE susceptible-infectious-susceptible compartmental model with potentially periodic behavior

“…The distinction of 2.2 in comparison to 2.1 is that it describes both the infected proportion and a time-varying average duration of infection through the product i ( t ) m ( t ), yet akin to 2.1, only the currently infected proportion contribute to infection, by means of the term β i ( x ) within the integral, (see (Greenhalgh and Rozins 2021) for further details). Imposing the assumption that , it follows that 2.2 reduces to the gSIS model (Greenhalgh and Rozins 2021): or given the conservation of population, s + i = 1, after a slight rearrangement, simply where , and .…”

“…Here, we propose a new take on the SIS model. We derive our model starting from the general formulation of SIS models as a system of integral equations (Brauer 2010; Greenhalgh and Rozins 2021) under a general assumption placed on the duration of infection distribution. For particular cases of this class of distributions, we show that our generalized susceptible-infected-susceptible model (gSIS) reduces to the traditional SIS model (Kermack and Mckendrick 1991a, b, c) with constant coefficients, the SI model (when the demographic turnover rate is set to zero), and the IR model, respectively.…”

To Simulate the Spread of Infectious Diseases by the Random Matrix

“…The result of simulations are shown in Fig. (16). It shows that wearing a mask can control the spreading of diseases, especially for the general infectious diseases the medical mask reduces the deaths to 0.…”

“…For examples, the SEIR model [10] (with S, E, I, and R the susceptible, exposed, infected, and recover respectively), the SIRS model [10] (with S, I, R, and S the susceptible, infectious, recovered, and susceptible respectively), and the model considers asymptomatic carriers [11] are proposed, where more classes or compartments according to the epidemiological status are considered. Beyond the deterministic compartment models, the stochastic compartment models are introduced to simulate the stochastic factors [9,[12][13][14] and other types of differential equations, such as the delay differential equation [10,15], nonlinear Volterra integral equations [16], and the fractional differential equation [14], are introduced into the compartment models. The differential equation based compartment models are indeed able to simulate various kinds of infectious diseases [17][18][19][20] and evaluate the effectiveness of the control measures [21,22] however, they ignore the individual differences [14], are unable to account for disease's true infectious period distribution [16], and are inflexible to simulate realistic scenarios such as random contact in the public places.…”

“…Beyond the deterministic compartment models, the stochastic compartment models are introduced to simulate the stochastic factors [9,[12][13][14] and other types of differential equations, such as the delay differential equation [10,15], nonlinear Volterra integral equations [16], and the fractional differential equation [14], are introduced into the compartment models. The differential equation based compartment models are indeed able to simulate various kinds of infectious diseases [17][18][19][20] and evaluate the effectiveness of the control measures [21,22] however, they ignore the individual differences [14], are unable to account for disease's true infectious period distribution [16], and are inflexible to simulate realistic scenarios such as random contact in the public places.…”