Impact of environmental transmission and contact rates on Covid-19 dynamics: A simulation study.

The emergence of the COVID-19 pandemic has been a major social and economic challenge globally. Infections from infected surfaces have been identified as drivers of Covid-19 transmission, but many epidemiological models do not include an environmental component to account for indirect transmission. We formulate a deterministic Covid-19 model with both direct and indirect transmissions. The computed basic reproduction number R 0 represents the average number of secondary direct human-to-human infections, and the average number of secondary indirect infections from the environment. Using Partial Rank Correlation Coefficient, we compute sensitivity indices of the basic reproductive number R 0 . As expected, the most significant parameter to reduce initial disease transmission is the natural death rate of pathogens in the environment. Variation of the basic reproduction number for different values of direct and indirect transmissions are numerically investigated. Decreasing the effective direct human-to-human contact rate and indirect transmission from human-to-environment will decrease the spread of the disease as R 0 decreases and vice versa. Since the effective contact rate often accounted for as a factor of the force of infection and other interventions measures such as treatment rate are prominent features of infectious diseases, we consider several functional forms of the incidence function, and numerically investigate their potential impact on the long-term dynamics of the disease. Simulations results revealed some differences for the time and infection to reach its peak. Thus, the choice of the functional form of the force of infection should mainly be influenced by the specifics of the prevention measures being implemented.

Introduction

The 2019 coronavirus disease (Covid-19) caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a global public health concern worldwide [1], with serious health, social, and economic consequences, a crisis of unprecedented proportions [2]. The disease spreads through direct transmission (person-to-person), which has been the focus of most modeling studies [3]. While human-to-human transmission dynamics is well established, little attention has been provided to indirect transmission of the virus through surfaces such as door handles, at least from the mathematical standpoint. It is well documented that the SARS-CoV-2 can persist on hard surfaces and in the environment [3], and as such, environmental factors may have a great impact on the transmission dynamics of the disease [4]. We assume that infectious individuals are responsible for shedding the virus in the environment, and susceptible individuals could be infected by accidental contacts with the virus on surfaces [5].

Several studies have included the environment in their mathematical model, but they only considered the contribution of infectives shedding the virus in the environment, that is the role human play by releasing the virus into the environment by including an additional compartment for contaminated environment [6], [7], [8], [9], [10], [11]. Specifically accounting for indirect human-to-environment transmission is important to investigate the impact this additional transmission route has on the long term dynamics of the disease. To address this gap, we include an additional force of infection to account for this indirect transmission [12], [13]. With increase in the number of cases despite the strict non-pharmaceutical (prevention) interventions (wearing face mask, practicing social/physical distancing, and environmental disinfection) suggests the importance of investigating the impact of indirect transmission of Covid-19 by coupling the transmission dynamic of the virus in the environment to the direct transmission modeling pathway. The proposed model is an improvement on the work in [12], [13], [14]. Given the current state of the pandemic, we include a vaccination class, with a vaccine waning rate. Motivated by the aforementioned, we develop a susceptible–vaccinated–exposed–infectious–recovered () deterministic mathematical model for the transmission dynamic of Covid-19 incorporating both direct (human-to-human) transmission, indirect (human–environment–human) transmission from the environment. Because the virus can survive several days on different surfaces, investigating the potential impact played by indirect Covid-19 transmission through contaminated environment could be crucial to mitigate factors that intensify the spread of the virus.

Infections from infected surfaces have been identified as drivers of Covid-19 transmission, and a lot of money have been spent on disinfecting surfaces and work places. Yet, many epidemiological models do not include an environmental component to account for indirect transmission. To study the potential impact of this indirect transmission on the Covid-19 transmission dynamics, we formulate and analyze a mathematical model that incorporates the environmental component. Preventive non-pharmaceutical control interventions have been accounted for in the disease force of infection, specifically by including these as part of the effective contact rate. Disease transmission/contact parameter is generally incorporated into the force of infection. To this effect, we numerically investigate different functional forms of the contact and treatment rates when the basic reproduction number is either less or greater than unity, and highlight what their effect could be on the long-term dynamics the disease. The aim is to answer the following question: Can the choice of the functional forms of the contact and treatment rates influence the long term dynamics of the disease? To address this question, we performed numerous numerical simulations varying key parameters.

The rest of the paper is organized as follows. The proposed model is formulated and the basic reproduction number computed in Section 2. Variations of the basic reproduction number and the infected class for different values of these parameters are shown graphically in Section 3, along with the long term dynamics of various effective contact rates on the infection dynamics. The conclusion is provided in Section 4.

The model

We developed a simplistic caricature -type compartmental model that combines direct and indirect Covid-19 transmission, including vaccination and the dynamics of contaminated surfaces (environment). Direct or indirect transmission requires contact between an animal source and humans or contact with the pathogen in the environment.

Consider a homogeneously mixing within the population, i.e., individuals in the population have equal probability of contact with each other. Using a deterministic compartmental modeling approach to describe the disease transmission dynamics, the total population at any time is subdivided into several epidemiological classes depending on individuals health status: susceptible , vaccinated , exposed , infectious , and recovered . The pathogen in the environment is denoted as (see Fig. 1). with initial conditions where

Fig. 1

Compartment diagram of the human component of the model.

, and , and .

We choose a Holling type-II functional response based on the fact that, even though the spread of Covid-19 is rapid, there is (will) always (be) a saturation point. The type-II functional response captures this saturation/inhibitory effect as the number of infectives becomes large [15]. The constant parameters measure this saturation effect. In the expression , we note that represents the half saturation constant (often denoted by ), that is the concentration of Corona virus on surfaces that yields a 50% chance of contracting Covid-19 [15]. All model parameters, their description, values and sources are provided in Table 1.

Table 1

Fundamental model parameter.

Parameter	Description	Value	Range	Reference
Π	Recruitment or inflow into the population	100059×365		[21], [22]
v	Vaccination rate	0.45	[10^−5,8×10^−2]	[23], [24]
w	Vaccine waning rate	0.0001		Assumed
p	Fraction of recruited who are susceptible	0.9
q	Fraction of recruited who are already vaccinated	0.1	[0,0.99]	[23], [24]
α_1	Proportion of interaction with an infectious individual	0.1		[12]
α_2	Proportion of interaction with an infectious environment	0.1		[12]
β_I	Rate of transmission from S to E due to contact with I	0.0115		[12]
β_P	Rate of transmission from S to E due to contact with P	0.00414		[12]
σ	Reduction of infectiousness of vaccinated	0.001		Assumed
μ	Natural death rate	159×365		[12], [21]
μ_P	Natural death rate of pathogens in the environment	0.1724		[12]
δ	Disease-induced death rate	0.0018		[12]
ϕ	Progression rate from E to I	0.09		[12]
τ	Recovery rate of infected individuals	0.05	[130,14]	[12], [25], [26]
η	Rate of virus spread to environment by infectious individuals	0.1	[120,110]	[12]

There are multiple epidemiological indicators used for discussing the spread of diseases such as the basic reproductive number, a threshold parameter which indicates whether the disease is going to spread or die out.

Disease-free equilibrium and basic reproduction number

The disease-free equilibrium of the system (1) is given by where

Using the next generation matrix method in [16], the associated next generation matrix and the rate of transfer of individuals of model system (1) are given by

Hence, the new infection terms and the remaining transfer terms are respectively given by

Thus,

The dominant eigenvalue or spectral radius of the next generation matrix which represents the basic reproductive number is given by where and

The expression of in (3) is made of two parts: the average number of secondary direct human-to-human infections, and the average number of secondary indirect infections from the environment.

The basic reproductive number denoted by which measures initial disease transmission is (a threshold value that characterizes the local asymptotic stability of the underlying dynamical system), and one of the most crucial quantities in infectious diseases [17]. It is defined as the average number of secondary infections generated by a single infectious individual over the duration of his infection, in an otherwise entirely susceptible/naive population [16]. The potential of an epidemic to spread very quickly or not is based on the classical behavior of being greater or less than unity [18]. Eq. (4) represents the infection from direct human-to-human contacts while Eq. (5) represents the infection contributed by indirect transmission through the environment. Similar expressions and interpretation of the basic reproduction number in Eq. (3) can be found in [19], [20].

Sensitivity analysis of

By construction, mathematical models (symbolic representations of real-life situation) inherit the loss of information. We perform sensitivity analysis by employing the Partial rank correlation coefficients (PRCCs) and the Latin hypercube scheme to identify model parameters that significantly impact disease transmission using the reproduction number as the response function. The sign of the PRCC indicates the specific qualitative relationship between the input parameter and . Parameters with large PRCC are strongly correlated with the basic reproduction number , that is, the larger the value of the PRCC, the larger is the contribution of the input parameter in the size of .

Parameter sensitivity analysis suggests that the most impactful parameters include the inflow of individuals into the community (the per capita recruitment rate) , the effective disease transmission/contact rates , the progression rate from exposed to infectious among others. As expected, that increasing the vaccination rate will decrease , while increasing the vaccine waning rate will increase the value of . Thus, early ranking the intervention measures based on their impact could ideally partially inform the process of prioritizing public health intervention measures to be implemented during the COVID-19. The PRCC analysis plot is depicted in Fig. 2.

Fig. 2

Partial rank correlation coefficients for simulations using the Latin hypercube sampling showing the impact of model parameters on the reproduction number .

Numerical simulations

Graphical representations depicting the dynamical behavior of the model system (1) when the fundamental threshold parameter is either greater or less than unity are presented. All model parameter values used in our simulations are shown in Table 1.

Heat map of for different values of the parameters considered

The variation of the basic reproduction number for different daily values of direct and indirect contact rates and is shown in the heat map in Fig. 3. Decreasing the effective direct human-to-human contact rate and indirect transmission from human-to-environment will decrease the spread of the disease as decreases and vice versa [27]. Similarly, the surface plot depicting the variation of for different values of direct transmission via the human-to-human contact rate and treatment rate is shown in Fig. 4. Over broad regions of parameter space, as the contact rate increases, also increases, and consequently, the treatment level to mitigate the spread of the epidemic via reduction of the number of secondary infections caused by an infective during his entire period of infection increases.

Fig. 3

Heat map of for different values of and .

Fig. 4

Heat map of for different values of and .

Fig. 5 is the health map of the variations of the daily contact and vaccination rates and . The more the contacts, the more the level of vaccination efforts necessary to control the epidemic. Fig. 6 depicts the impact of the vaccination waning immunity of the dynamics of the disease.

Fig. 5

Heat map of varying the value of direct contact and vaccination rates and .

Fig. 6

Heat map of for different values of vaccination and waining immunity rates and .

Long term dynamics of the infected component with and

Case 1:Impact of indirect transmission - The impact of indirect (environmental) transmission on the Covid-19 infection dynamics is graphically represented for with and without direct environmental transmission (i.e., the environmental-induced force of infection is zero or not) when the basic reproduction number is either greater or less than unity. As expected, from Fig. 8, the introduction of environmental perturbations (transmission) has a clear impact on the disease dynamics as this increases the number of infected individuals [28]. Therefore, additional mitigation strategies should be put in place such as regularly disinfecting surfaces and hand sanitization (see Fig. 7).

Fig. 8

Dynamics of infected individuals with and without indirect transmission when .

Fig. 7

Dynamics of infected individuals with and without indirect transmission when .

An epidemic is a function of environmental factors and a contact structure that varies over time, which in turn leads to varying transmission potential of an infection. Different forms of the transmission rates , from constant to dynamic contact rate abound in the literature. We numerically investigate the impact of (some of) these contact rates on the long term dynamics of the disease. These are graphically represented below for the infective class only.

Case 2:Face masks vs. isolation of infectives and lock down - While face masks have been highly successful in mitigating the spread of Covid-19, it has been noticed that fatigue is creeping in. We compare the impact of face masks and a combination of isolation of infectives and complete lock down. The latter prevention measure has very high economic consequences on people’s daily life. Efficacy of mask usage at reducing the risk of direct transmission of Covid-19 has been included in the contact rate [29], [30]. In this case the effective contact rate , with , where is the efficacy of face masks, is the proportion of those using face masks. If lock down and isolation of infectives are implemented in the community, these are accounted for in the model as part of the effective contact rate , with  [31]. The parameter represents the lock down rate of susceptible, while is the isolation rate of infectives. Although Fig. 9, Fig. 10 are similar, the value of the reproduction number are slightly different as indicated in the figures’ captions. This indicates that adhering and consistently wearing face masks somehow have a similar effect as when isolation of infectives and the stringent lock down measure are implemented.

Fig. 9

Dynamics of infected individuals with when and .

Fig. 10

Dynamics of infected individuals with when and .

The figures below are generated with and .

Case 3:Social/physical distancing, hand washing and face mask vs. time dependent contact rate - Riyapan et al. [29] introduce the effect of physical distancing, hand washing and face masks wearing , with , where is the proportion wearing face mask, is the efficacy of the face mask, the proportion practicing physical distancing, is the proportion implementing hand washing. It has been shown that is particularly sensitive to social/physical distancing and hand washing in this case [32]. On the other hand, assume  [33], where is the initial contact rate, is the final contact rate, and is the exponential decreasing contact rate. The dynamics of the infective class in both cases depicted in Fig. 11, Fig. 12 are somewhat similar, but the values of are different.

Fig. 11

Dynamics of infected individuals with .

Fig. 12

Dynamics of infected individuals with .

Case 4:Convex contact rate and seasonal variation - We consider the following convex contact rate , and  [34], where represents the proportion of individuals protected from environmental transmission [3], that is those practicing hand washing and disinfection of surfaces. Fig. 13 depicts this case. While the shape of the epidemic is similar to cases 1 and 2 above, we observe a little hump around day 125 of the epidemic when , indicating that even though the epidemic has almost reach its disease-free equilibrium, a second wave could emerge. This is in line with the current dynamics of the Covid-19 pandemic as several countries are experiencing a second or third wave, while there was a believe that the disease was under control (due to strict lockdown measure after the first wave).

Fig. 13

Dynamics of infected individuals with and .

Effective disease contact rates are often driven by environmental factors such as seasonal variations. The seasonality is widely used in epidemic models, and it can make the system more complex, where nonlinear dynamics are found in the system. Even though the Covid-19 pandemic has been around for less than two years and seasonal variations are yet to be observed [35], we investigated the potential effect of seasonality on infection dynamics from the mathematical standpoint. Consider a contact rate including a function, commonly used to model periodic phenomena such as influenza-like epidemics  [15], [36]. Fig. 14 shows as expected that the resonance effect tapers off with time, that is, the disease will taper off after about 3 years, indicating that the world might still be a long way off to see the end of this pandemic.

Fig. 14

Dynamics of infected individuals with .

Case 5:Recovery and natural death rates depend on the infections fatality rate vs recovery and natural death rate are time dependent functions - When the recovery and natural death rates depend on the infections fatality rate (CFR), these are respectively given by , and , where (unit of are days [37]. Fig. 15 depict this case. On the other hand, Fig. 16 is the graphical representation of the case when recovery and natural death rates are time dependent functions given by , and , with  [38]. One notes that the infections dies off quicker when case fatality rate is accounted for in Fig. 15 compared to when recovery and natural death rates are time dependent as shown in Fig. 16.

Fig. 15

Dynamics of infected individuals with and .

Fig. 16

Dynamics of infected individuals with and .

Finally, Covid-19 has affected healthcare resources (e.g., number of hospital beds) [39]. Thus, recovery rate could be considered as a linear incidence function which depends on the number of hospital beds, that is, , where is the hospital-bed to population ratio [40], represents the maximum removal rate when the number of hospital beds sufficient, represents the minimum per capita removal rate that can be sustained in the face of a large number of infections [40], [41], [42]. Graphical representation of this case is similar to Fig. 15, Fig. 16 and for this reason is not represented here.

Conclusion

The novel and highly contagious coronavirus disease has rapidly spread across the globe, causing unprecedented socio-economic burden on communities. Several modeling studies have accounted for direct transmission in their mechanistic models. Herein, we formulated a deterministic Covid-19 model with both direct and indirect transmissions, and computed the basic reproduction number. The impact of indirect (human-to-environment) transmission is investigated, and results indicate that accounting for environmental perturbations has a clear impact on the disease dynamics as there are more infections than when indirect transmission is ignored. Consequently, additional strategies to mitigate the spread of the virus on surfaces such as disinfection will be required.

Next, because non-pharmaceutical control interventions are incorporated in several studies as a factor of the force of infection, we numerically investigate the effect of some of these functional forms on the contact and treatment rates on the disease spread when the basic reproduction number is either less or greater than unity. While the shape of most these graphs are similar (using the same model parameter values), key differences are the peak of the infection and the time to reach this peak, as well as the time when the hump tapers off.

One question was posed in this study. To determine whether the choice of the functional forms of the contact and treatment rates influence the long term dynamics of the disease. To better answer this question, we considered several forms of the contact rate. Numerical simulations were then carried out to investigate their impact on the disease dynamics. Results based on functional forms of the contact rates show some differences and could even substitute each other. Thus, the choice of the functional form of the effective contact rate has an impact on the dynamics of the infection. This highlights the importance of carefully accounting for the control measures being implemented prior to investigating their impact on the disease dynamics, as the choice and form of the contact rate should be specific to the prevention measures being implemented.

This study to qualitatively assessing via numerical simulations the impact of direct and indirect transmissions as well as the role played by the various forms of the contact and treatment/recovery rates is not exhaustive. A literature review of additional functional forms used in the literature could be conducted and compared as done herein. Early estimation and calibration of the model with observed (daily) epidemiological data is critical to provide data-driven information to public health and policy decision-makers at the onset of the epidemic/pandemic.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.