key: cord-0273205-2yyvs2bs authors: Ghosh, A.; Mubayi, A.; Pandey, A.; Brasic, C.; Ghosh, P. title: Estimating Time-Dependent Disease Transmission Intensity using Reported Data: An Application to Ebola and Selected Public Health Problems date: 2021-06-07 journal: nan DOI: 10.1101/2021.06.04.21258380 sha: 65ad2361764895be5c8e4061918ef6cebe9ad2ee doc_id: 273205 cord_uid: 2yyvs2bs Obtaining reasonable estimates for transmission rates from observed data is a challenge when using mathematical models to study the dynamics of infectious diseases, like Ebola. Most models assume the transmission rate of a contagion does not vary over time. However, these rates do vary during an epidemic due to environmental conditions, social behaviors, and public-health interventions deployed to control the disease. Therefore, obtaining time-dependent rates can aid in understanding the progression of disease through a population. We derive an analytical expression using a standard SIR-type mathematical model to compute time-dependent transmission rate estimates for an epidemic in terms of either incidence or prevalence type available data. We illustrate applicability of our method by applying data on various public health problems, including infectious diseases (Ebola, SARS, and Leishmaniasis) and social issues (obesity and alcohol drinking) to compute transmission rates over time. We show that transmission rate estimates can have a large variation over time, depending on the type of available data and other epidemiological parameters. Time-dependent estimation of transmission rates captures the dynamics of the problem and can be utilized to understand disease progression through population accurately. Alternatively, constant estimations may provide unacceptable results that could have major public health consequences. An epidemic is a function of environmental factors and a contact structure that 20 varies over time, which in turn leads to varying transmission potential of an "infection". 21 We also refer the word "infection" to describe social influences exerted by a typical 22 influential individual with a particular social problem that results in a naive (to the social 23 problem) individual getting involved in the problem. For example, an alcoholic might 24 influence an abstainer into initiating drinking. Many authors have studied outbreaks 25 of social problems and infectious diseases using compartmental transmission model. 26 Qualitative aspects of homogeneous mixing models with constant transmission potential 27 of an infection are well understood for various applications. These models are relatively 28 easy to analyze and can answer questions, at the population level, with good precision. 29 which includes the contact rate c, is called as a "transmission coefficient" (or "effective 48 contact rate" or "transmission potential:) with units as time −1 . At low population 49 densities mass action is a reasonable approximation of a much more complex contact 50 structure, however, in general, standard incidence is more appropriate for modeling 51 transmission for human diseases or influences for social problems. The term βI/N is individual. The transmission rate is calculated by dividing incidence for a given time 56 period by a disease prevalence for the same time interval. 57 Most infectious disease data is collected in form of incidence and/or prevalence. In compartmental mathematical models, varied assumptions are made based on 75 characteristics of a modeling disease which lead modelers to focus on more important 76 aspects of the epidemic. For example, an epidemic that occurs on a timescale that is 77 much shorter than that of the population replenishment (that is, epidemic occurs at a 78 much faster rate than births and deaths in the population), constant population size can 79 be assumed. Additional common features of these models might include temporary or 80 permanent recovery of infected individuals and a birth rate into infective class. Whether 81 establishment or a major outbreak of an infectious disease or a social problem will 82 occur in a population, requires extensive experience or a mathematical model of disease 83 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 7, 2021. In this paper, we compute time dependent and independent transmission coeffi- . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 7, 2021. Consider a "disease" outbreak in a population that follow the following system of 118 differential equations: (1) where R(t) = 1 − S(t) − I(t) and parameters are defined in Table 1 and Table 2 . Transmission or influence coefficient Per-capita recovery rate γ(t) Per-capita rate of loosing immunity or relapse rate µ(t) Per-capita mortality or departure rate Setting c(t) = b(t) + γ(t) and d(t) = γ(t) + µ(t) in Equation ( . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 7, 2021. ; https://doi.org/10.1101/2021.06.04.21258380 doi: medRxiv preprint Isolating β(t) from Equation (2) we obtain β(t) as function of prevalence (I) where S(t) is given by Equation (4). Note, beside prevalence (I), we also need I to compute β(t) using formula 5. However, I can be approximated using prevalence data. On the other hand, suppose incidence data are available. In order to calculate 132 expression of β(t) as a function of incidence (w(t) = β(t)SI) we first solve Equation (2) for I with initial condition I(T) (where T ∈ [0, L] is a time at which the prevalence Thus, where S(t) and I(t) are given by Equations (7) and (6), respectively. Note, we need prevalence at time point T, I(T), to compute β(t) using formula (7). The time point T can be appropriately chosen, close to maximum of prevalence and not 142 towards starting or end of epidemic. Let θ represents vector of our transmission parameters and y = (y 1 , y 2 , ....., y T ) T is the available data set. We can take likelihood function in our bayesian approach as where T is the total number of data points in the data set, σ is the appropriately chosen We use four examples to show how to estimate β over time from the available epidemiological data. The examples provide a method to study social and public health issues. To compute estimates of β(t), we use first order discretization for derivatives and composite trapezoidal rule for integration as given below These discretization are used in the formulas given in Equations (5) and (8). 158 We can avoid this discretization by choosing a function, for example, a polynomial, 159 that can be fitted to the prevalence and incidence temporal data. This fitted function can 160 then be used directly in the Equations (5) and (8). In this section, we apply available incidence data to three past epidemics: the . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted June 7, 2021. ; https://doi.org/10.1101/2021.06.04.21258380 doi: medRxiv preprint On discretizing Equation (10) we get following expressions. If t ≤ T, For the estimation of β(t) with regards to available incidence data, the estimates 182 are found in Table 2 median β(t) to be that of Guinea (see Table 3 and Figure 2 ). Analysing the estimates for 186 transmission rate temporally, we observe that transmission rate follows the incidence . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted June 7, 2021. ; https://doi.org/10.1101/2021.06.04.21258380 doi: medRxiv preprint . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted June 7, 2021. The first model was for a single outbreak and hence demography was not considered 198 whereas the second model assumed birth and death though with a same per-captia rate. If t ≤ T then where where where g i (x) = e m i x w(x) (for i = 1, 2), m 2 = µ and m 3 = α + µ. The obtained estimates of β(t) are given in . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. On discretizing Equation (15) we get following expressions. If t ≤ T, . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted June 7, 2021. ; https://doi.org/10.1101/2021.06.04.21258380 doi: medRxiv preprint If t > T, where g 4 (x) = e m 4 x w(x) and m 4 = α. 226 The temporal estimates of β(t) are shown in Table 4 (see Appendix A.2) and Figure 227 7b. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted June 7, 2021. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted June 7, 2021. The recovery rate, α is taken to be 0.17 [4] . We estimate β(t) using simplified Equation (5) and above assumptions as follows where If µ = 0, this equation can be reduced, where f 2 is −αI(x). 250 We found that mean estimate of β is 1.04 (std=0.3; Table 5 Table 6 (see Appendix A.2) with range of (0.36, 3.02). This is because the region of 288 our study differ from the region modeled by [17] . Our results suggest that estimates 289 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted June 7, 2021. where and The following abbreviations are used in this manuscript: Prevalence data 436 We demonstrate our method of using prevalence data to estimate time dependent 437 transmission coefficient using synthetic prevalence data generated with two particular 438 choices of transmission coefficients (constant and seasonal with respect to time) and the 439 model ((1) and (2)) with rest of parameters given by Table A1 . Table A1 . Parameters for generating synthetic prevalence data. Values 1 33/1000 0 33/1000 0 We used two particular choice of transmission coefficients, is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted June 7, 2021. ; https://doi.org/10.1101/2021.06.04.21258380 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 7, 2021. ; https://doi.org/10.1101/2021.06.04.21258380 doi: medRxiv preprint Table 5 . Alcohol drinking data and estimates of β(t). Drinkers (I(t)) 52% 51% 49% 46% 47% β(t) when µ = 0 − 0.85 0.84 0.90 1.54 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 7, 2021. ; https://doi.org/10.1101/2021.06.04.21258380 doi: medRxiv preprint . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 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