Analytical solution for post-death transmission model of Ebola epidemics

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Abstract

In this paper, we shall introduce deceased human (D) transmission to the cycle phenomenon of disease modelling, which has a direct relationship with the infective compartment of the stochastic Susceptible-Infected-Recovered (SIR) disease model. Due to this, the noise covariance matrices of the standard stochastic SIR model will be modified. This will be done by using van Kampen's expansion method to approximate the master equation and the stochastic Fokker–Planck equation to analytically calculate a power spectral density (PSD) expression. A vector-valued process is used and shows that the absolute value of the real part of the principal diagonal of the PSD matrix solution is the spectral density of the system which is compared to the average PSD of the stochastic simulations. We aim to investigate the problem of identifiability when deceased humans act as an extended state of host infection using the SIR-D model during Ebola epidemics. Using our analytical solution model, we show that for an increasing degree of transmission parameter values the infected route cannot be identified, whereas the deceased human transmission shows enhancement for the persistence of Ebola virus disease using epidemiology data of the Democratic Republic of Congo.

Introduction

Ebola virus disease (EVD) is one of the most deadly diseases in Africa. It was first discovered in 1976 in the Democratic Republic of Congo (DRC), previously known as Zaire, and has a potential fatality rate of up to 90% [1], [2]. While the virus has been found in other parts of the world, the majority of cases and outbreaks are African [3]. Since 1976, Africa countries like Sudan, Gabon, Uganda and the DRC have experienced more than three outbreaks each [4]. The DRC, in particular has experienced 10 outbreaks [3] and the most recent outbreak was still ongoing during the writing this paper. As of 10 September 2019, there have been 3091 reported cases and 2074 fatalities [5]. There are six strains of the virus: Zaire, Sudan, Tai Forest, Bundibugyo, Reston and Bombali Ebola virus. Of these, only the first four are known to cause disease in humans [3], [6]. The Reston virus can cause disease in nonhuman primates [6] and Bombali Ebola virus, which was discovered in Sierra Leone in 2018, has not yet been confirmed as causing disease in either animals or humans [3]. Sierra Leone has recorded the highest number of confirmed cases (8706) and fatalities (3956) during the 2014–2016 EVD outbreak in West Africa [3].

However, post-death transmission plays an active role in infection transmission, causing susceptible populations to become infected either through a living host or a host that died with the virus [6], [7], [8], [9], [10]. Almost all initial cases of EVD have resulted from spillover events [10], [11], when an uninfected individual contacted an infected dead animal, such as a fruit bat or nonhuman primate [3], [12], [13]. The transmission process then continued with human-to-human contact. The persistence of the virus in a community has been decisively linked by scholars to funeral practices [14], [15]. Direct contact with the blood or bodily fluid of an infected individual showing signs and symptoms of the virus (e.g., vomit, high fever, diarrhoea, generalised pains) can also cause transmission. Since the virus can survive outside of a living host for several days [4], [16], [17], it may spread via contaminated objects. There is no confirmed source indicating where the virus originated, but various studies have proven that the African fruit bat is a known animal reservoir for the virus [12], [13]. Although trial vaccinations and drugs were employed in the 2018 DRC outbreaks, there is no licenced vaccine or medication for the virus at the time of writing.

Mathematical models have been used to study the dynamics of the outbreaks with the deceased human acting as an extended state of host infection using the Susceptible-Exposed-Infectious-Recovered-Deceased (SEIRD) type model [8], [18], [19] and its extension [2], [10], [20], [21], [22], [23], [24], [25]. Although an analytical application can give a better understanding of an epidemic model using a nonlinear differential equation [26], numerical simulations are the most common computer application considered to study the dynamics of the disease with these models. Meakin et al. [25] used a numerical simulation on a simple stochastic metapopulation model to study the dynamics of the 2018 DRC Ebola outbreak with time-series data for three health care centres in Equateur province. Do and Lee [18] provided analytical and numerical analyses of EVD dynamics without vital dynamics (that is, the birth and death processes neglected) using the SLIRD model (L is latently individual) to identify sensitive transmission areas in Nigeria during 2014–2016 West Africa outbreak. They reported that safe burial was a targeted area that required effective intervention strategies to combat the spread of the disease. Their findings were reinforced by Salem and Smith [19], who used the SEIRD model and provided a sensitivity analysis with a Partial Ranking Correlation Coefficient (PRCC) and Latin hypercube sampling. The major difference between these two studies is that, in [18], transmission by latently infected deceased individuals was included in the model and for Salem and Smith [19], the model was deterministically analysed with vital dynamics. Also, Agusto et al. [27] used PRCC to assess the impact of population-level on the basis of non-pharmaceutical control measures during the 2014 EVD outbreak in West Africa by incorporating the effects of traditional belief systems and customs, along with disease transmission within health-care settings and by Ebola-deceased individuals. The authors reported that in the absence of anti-Ebola public health interventions, traditional/cultural/custom belief systems through which the interaction between an uninfected individual and Ebola-deceased individuals is very high [14] is one parameter that has the most influence on the Ebola transmission dynamics in Guinea.

In almost all of these studies, the importance of post-death transmission route has been understood using the basic reproduction number or sensitive parameters as a response function. The basic reproduction number is a parametric equation used to determine whether a disease will persist or not [27]. Weitz and Dushoff [8] reported that there was an identifiability problem when the SEIR was modelled with post-death transmission using epidemic growth rate data with the infective equation as a response function. This was accomplished by comparing the growth rate of the SEIR and SEIRD models using numerical simulation. It was shown that, for the latter, different transmission parameter values produced almost the same pattern of epidemic curves, indicating challenges in understanding the dynamics of the disease, as epidemiology parameters for EVD are difficult to identify. This motivated us to investigate the dynamics of the disease using the cycle phenomenon, for by using the Fokker–Planck equation, the infective equation can obtain a Gaussian probability distribution curve, whose peaks position change to a higher frequency as the transmission parameter values increases [28], [29]. That is, in contrast to Yamazaki [30] where a diffusive term was added as an extension to a host-deceased-pathogen model to explain the stabilities for the movement of human host for EVD dynamics, we are extending a host-deceased model by adding a fluctuating term into the deterministic mean-field equation and analysing the dynamics of EVD in the form of cycles. A model cannot be identified if for different degree of the transmission parameters, the infective equation produce oscillating curves which do not change their peak positions [28] or produce a flat peak that covers a wide frequency range, indicating that the infective curves cannot provide a better description of the epidemic dynamics [31]. Notwithstanding this, stochastic applications with deceased transmission models are less common with vital dynamics. Nieddu et al. [10] considered it, but used the Monte Carlo algorithm to generate the stochastic simulation to explain the dynamics of the disease.

Stochastic Susceptible-Infected-Recovered (SIR) models with vital dynamics have been considered analytically using different techniques without the deceased compartment. In [32], an exact analytical solution was presented showing how the SIR model can be reduced to an Abel equation by using a perturbative approach in a power series form. Also, to understand the interplay between the deterministic and stochastic model and also to give a better description of the dynamics of an epidemic, an analytic power spectral density (PSD) has been considered. Alonso et al. [33] presented an analytic PSD model to investigate stochastic amplification in an epidemic model, considering the internal and external transmission. It was shown that demographic noise could shift the damped oscillation of the deterministic period, and the oscillating infective curve of the PSD expression moved to a higher frequency as the transmission value increased. Moreover, a stochastic SIR model with seasonal forcing was also used with analytic PSD techniques in [34] to separate the connection between external forcing and stochasticity in an epidemic model. Also, to investigate the performance of the resonant fluctuation of the stochastic SIR model in [33], an analytical PSD solution was presented by relaxing the random mixing assumption and including a mixing network by Simões et al. [35]. The analytic PSD solution has also been considered in [26], [29], [36], [37], [38] on epidemic dynamics.

All of these works used a single species model except [29], where a two-species model was used analytically to obtain a PSD expression. In this model [29], there is no direct relationship between the infectious state and the other transmission state (environment), as would be in the case with EVD, where death transmission is considered. This scenario makes our model unique for its analytic prediction model for epidemic dynamics modelling. In this study, we aim to investigate the EVD epidemic curve pattern analytically when deceased humans act as an extended state of host infection given the persistence of the EVD outbreak in DRC. This is performed by developing a fully stochastic model with living-to-dead and living-to-living transmission, through contact between individuals within the human population, using the SIRD model. We derive an analytic PSD expression from a stochastic Fokker–Planck equation of the number of susceptible, infected and deceased individuals. This method helps us to analytically describe the fluctuation produced by demographic stochasticity [29], [33] during an EVD outbreak. Our analysis sheds new light on the identifiability problem on the dynamics of EVD when a deceased human is a transition point of host infection. Our results show that, by controlling the post-death transmission route, it is possible to bring the frequency of the EVD outbreak to a weaker state.

The organisation of this paper is as follows. In Section 2, we present the full description of the stochastic model and its formation, showing that approximating the master equation using one step operator techniques we can have an equation which mathematically it is equivalent to a set of a linear stochastic differential equation. The endemic steady state of the system investigated, and we use the next generator approach to determine the basic reproduction number. In Section 3, we provide the analytic prediction calculation (PSD calculation) and analyses of both numerical and analytic findings in Section 4. The conclusion of the paper is given in Section 5.

Section snippets

The stochastic SIR model with a deceased transmission

Although the SEIR type model is more appropriate to investigate the dynamics of EVD [4], [8], [10], we have limited our analysis to four classes of SIRD model. The susceptible class (S) consists of individuals who can become infected with the virus by having contact with an individual in the infectious class (I) or deceased class (D). The D class is for those individuals that died with the virus but not yet buried [4], [8] while the recovered class (R) is for those who have recovered from the

The spectral density calculation for the stochastic SIR-D

The steady-state solution of the stochastic SIR-D model is studied with a Fourier analysis method using spectral density which is the squared absolute value of the Fourier transform process. Since the spectral density of the white noise process is a matrix-valued function, we calculate the PSD using vector-valued processes. However, because the endemic state of system is stable therefore the stochastic model can be represented in the form of a Langevin equation defined asdy(t)dt=Ay(t)+LW(t),

Numerical simulations for the stochastic and deterministic SIR-D model

We first investigate that the endemicity of the deterministic model produced an oscillation which corresponded to the stochastic model that fades away with time using Eq. (8). The trajectory for the stochastic model performed numerically using the Gillespie stochastic simulation algorithm [43] using R-software. We demonstrated the relationship between the stochastic and deterministic simulations of the deceased and infectious states using the data in Table 2.

The GillespieSSA package [44] was

Conclusion

In this paper, we presented a full stochastic host-deceased model of an EVD outbreak with two routes of transmission (living-to-dead and living-to-living) using the SIR-D model. We applied the method of van Kampen's expansion [40] to approximate the master equation of the stochastic model. The solution of this method yields a nonlinear multivariate Fokker–Planck equation, which is equivalent to a Langevin equation. An analytically derived PSD expression of the number of susceptible, infected

Funding

This work supported by the Key Project of National Science Foundation of China (11531006).

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