1 Introduction

Airborne transmission of infectious diseases has been a subject of increasing interest in recent times. Analyzing the potential risk of airborne transmission by incorporating human mobility into studies can provide important information for designing safe environments with appropriate level of controls for mitigating risks. Many airborne and waterborne diseases are transmitted directly via host–host and/or indirectly through the host–source–host route (Boone and Gerba 2007; Trisha et al. 2021). Airborne diseases such as measles, influenza, and tuberculosis are primarily transmitted from human–environment–human (indirect transmission pathway) and can as well be transmitted from human–human (direct transmission pathway) (Hartley et al. 2006; Nelson et al. 2009; Noakes and Sleigh 2009). The emergence of severe acute respiratory syndrome (SARS) in 2002–2003 caused a public health concern (Steven et al. 2003)with a highly infectious coronavirus causing the SARS disease (Lipsitch et al. 2003) known to primarily spread through localized contact with contaminated droplets. In addition, retrospective studies including but not limited to Li et al. (2005) suggested that airborne dispersal may play an important role in the transmission of the disease with evidence showing that uninfected individuals were infected without enough close contact with an infectious individual (Scales Damon et al. 2003).

In the context of models for infectious diseases transmission, the Lagrangian approach offers a theoretical framework that explicitly tracks the heterogeneity of host population and mobility (Fred et al. 2019). Despite the complexity of explicitly considering the mixing patterns and connections that exist between local communities and different regions, the Lagrangian method used for residence time has shown its ability to analytically explain the failure of feasible mobility restriction measures and their impact on disease transmission (Brauer 2012; Fred et al. 2019). Several mathematical models that studied heterogeneity and different mobility scenarios have been explored within the heterogeneous mixing framework (Brauer 2008b, 2012; Fred et al. 2019). However, none of these studies considered epidemic models with diffusion of pathogens and/or indirect transmission route using our approach of a coupled PDE–ODE model.

In fact, many authors including but not limited to the authors in David et al. (2020), Noakes and Sleigh (2009), Wang and Wang (2021), Yamazaki et al. (2021) and Zhang et al. (2016) extensively studied and analyzed the transmission of airborne and waterborne diseases using different modeling approaches. For example, (Noakes and Sleigh 2009) used a stochastic version of the Wells–Riley’s model coupled with a simple zonal ventilation model that allowed for the effects of small populations applicable in healthcare settings. They demonstrated the role of airflow and population size on the risk of infection and the importance of stochastic effects, especially in small populations, was emphasized. Their approach enabled an understanding of the possible spatial transmission of infection that allowed design and operational control strategies to be explored without explicitly modeling the role of diffusion. Wang and Wang (2021) developed a multipatch cholera epidemic model to assess disease dynamics in a periodic environment. They illustrated the impact of asymptomatic infections and population dispersal on the spread of cholera. Although useful results were obtained, their modeling framework did not show the effect of diffusion and human mobility on the transmission of the disease. Similarly, (Yamazaki et al. 2021) proposed a new reaction–convection–diffusion model to study the impact of cholera transmission among humans using a second-order differential operator. This model differs from the current study in structure and dynamics in that a second-order differential operator was used without addressing the potential effect of the movement of humans between patches. Furthermore, (Zhang et al. 2016) formulated a reaction–diffusion waterborne pathogen model to investigate the influence of diffusion, spatial heterogeneity, and multiple transmission pathways on the spread of waterborne disease. While the work done in Zhang et al. (2016) considered the influence of dispersal using a different model, the spatiotemporal spread of the disease with the impact of human mobility was not taken into consideration. In addition, the work of David et al. (2020) which is a prequel to the current study used a novel PDE–ODE model to study the spatial spread of airborne diseases between homogeneous populations. The current work extends the work done in David et al. (2020) by explicitly modeling the role of human mobility.

In light of the aforementioned studies, we thought it could be possible to consider a more realistic but simpler scenario (in the case of superspreaders) rather than the detailed network or PDE models. To model the effect of human mobility with heterogeneous mixing, one may assume that the population is divided into subgroups with different activity levels. To the best of our knowledge, no previous studies and articles considered our modeling approach of using a coupled PDE–ODE system in assessing the role of human mobility and heterogeneous mixing on the transmission of airborne diseases. This work aims to indirectly overcome the complications faced by using a PDE model. We base our analytical results on the reproduction number and the epidemic final size relation in a heterogeneous mixing environment. Our results are computed using a nonlinear system of ODEs derived from the coupled PDE–ODE using strong localized perturbation theory in the limit of fast diffusing pathogens. We use the Lagrangian method to track an individual’s place of residence at all time following the approach in Bichara et al. (2015), Brauer (2008b), Fred et al. (2019), Carlos et al. (2016), Espinoza et al. (2016) and Funke (2018). This allows us to theoretically and numerically evaluate how human movement between patches (regions) affect the spread of diseases. Our extended model which includes human mobility and a diffusion term may be an alternative way to study the spread of airborne diseases in an heterogeneous mixing environment.

The rest of the paper is structured as follows. In Sect. 2, we extend the coupled PDE–ODE model developed in David et al. (2020) by incorporating human mobility using the Lagrangian method. We non-dimensionalize the extended model in Sect. 2.1 and use matched asymptotic expansions method to reduce the dimensionless coupled model to a nonlinear system of ODEs in the limit of fast diffusing pathogens. In Sects. 3.1 and 3.2, for a scenario with two populations, the reduced ODE model is used to compute the basic reproduction number and final size relation of the epidemic, respectively. The coupled PDE–ODE model together with the reduced system of ODEs is used to numerically study the effect of different human mobility patterns on the disease dynamics for a two-patch scenario. A brief discussion of results and possible future directions concludes the paper in Sect. 4.

2 Mathematical Model

Here, we represent human populations with localized patches that have partially transmitting boundaries through which pathogens are shed into the atmosphere by infected individuals. These pathogens diffuse and decay at a constant rate in the air. A susceptible individual becomes infected by coming in contact with pathogens (indirect transmission pathway). We assume that the spread of infection in each population patch depends on the density of pathogens around the patch, and did not explicitly model pathogens within the patches. Similarly, we model human movement between patches using a Lagrangian method. Let \(\Omega \subset {\mathbb {R}}^2\) be our 2-D bounded domain of interest containing m population patches represented, by \(\Omega _j\) for \(j=1,\dots ,m\), which are separated by an \({\mathcal {O}}(1)\) distance. In the region between the patches \(\Omega \setminus \cup _{j=1}^{m}\,\Omega _j\) (bulk region), the density of pathogens \({\mathcal {P}}(\pmb {X},T)\) satisfies

$$\begin{aligned} \frac{\partial {\mathcal {P}}}{\partial T} =&\, D_B \, \Delta \, {\mathcal {P}} - \delta \, {\mathcal {P}}, \quad T>0,\quad \pmb {X} \in \Omega \setminus \cup _{j=1}^{m}\,\Omega _j; \end{aligned}$$
(1a)
$$\begin{aligned} \, \partial _{n_{\pmb {X}}} \, {\mathcal {P}} = \, 0, \quad&\pmb {X} \in \partial \Omega ; \qquad D_B \, \partial _{n_{\pmb {X} j}} \, {\mathcal {P}} = -r_j \, {\mathcal {I}}_j, \quad \pmb {X} \in \partial \Omega _j,\qquad j = 1, \dots , m, \end{aligned}$$
(1b)

where \(D_B>0\) and \(\delta > 0\), respectively, represent the dimensional diffusion coefficient and decay rate of pathogens in the bulk region, \(r_j>0\) is the dimensional shedding rate of pathogen by an infected individual in the \(j{\text {th}}\) patch, and \(\partial _{n_{\pmb {X}}}\) is the outer normal derivative on the boundary of the patches pointing into the bulk region.

First, we assume that the population of the entire system is constant throughout the epidemic period, that is, \(N = \sum _{j=1}^{m} N_j\). Here, \(\gamma _{ji}\) denote the fraction of residents of patch j that are in patch i, where \(\sum _{i=1}^{m} \gamma _{ji} = 1\) for \(j=1,\dots ,m\). The parameters \(\mu _j\) and \(\alpha _j\) are the dimensional transmission and recovery rates, respectively, for individuals in the \(j{\text {th}}\) patch, and \(p_c\) is a typical value for the density of pathogens. The dynamics of the diffusing pathogens in the bulk region is coupled to the population dynamics of the \(j{\text {th}}\) patch as follows

$$\begin{aligned} \frac{\text {d} {\mathcal {S}}_j}{\text {d}T}&= - \sum _{i =1}^{m} \mu _i \, {\mathcal {S}}_j\, \gamma _{ji} \int _{\partial \Omega _i}\, ({\mathcal {P}}/p_c) \,\,\,\text {d}S_{\pmb {X} i}; \end{aligned}$$
(1c)
$$\begin{aligned} \frac{\text {d} {\mathcal {I}}_j}{\text {d}T}&= \sum _{i =1}^{m} \mu _i\, {\mathcal {S}}_j\, \gamma _{ji} \int _{\partial \Omega _i}\, ({\mathcal {P}}/p_c) \,\,\,\text {d}S_{\pmb {X} i} - \alpha _j {\mathcal {I}}_j; \end{aligned}$$
(1d)
$$\begin{aligned} \frac{\text {d} {\mathcal {R}}_j}{\text {d}T}&= \alpha _j {\mathcal {I}}_j,\qquad j = 1, \dots , m, \end{aligned}$$
(1e)

where \({\mathcal {S}}_j, \, {\mathcal {I}}_j\), and \({\mathcal {R}}_j\) are the susceptible, infected, and recovered/removed population in the \(j{\text {th}}\) patch, respectively, with the total population at time T written as \({\mathcal {N}}_j(T) = {\mathcal {S}}_j(T) + {\mathcal {I}}_j(T) + {\mathcal {R}}_j(T)\).

The integrals in (1c) and (1d) are over the boundary of the \(j{\text {th}}\) patch and are used to account for the total density of pathogens around the patch. These terms show that the spread of infection within a patch depends on the density of pathogens around the patch. It is important to emphasize that our model does not explicitly account for pathogens within the patches. The Robin boundary condition \(D_B \, \partial _{n_{\pmb {X}}} \, {\mathcal {P}} = -r_j \, {\mathcal {I}}_j\) on the boundary of the \(j{\text {th}}\) patch accounts for the amount of pathogens shed into the atmosphere by infected individuals in the patch. This condition shows that the amount of pathogens shed into the atmosphere from the \(j{\text {th}}\) patch depends on the population of infected individuals in the patch.

2.1 Non-dimensionalization of the Coupled PDE–ODE Model

Next, we non-dimensionalize the coupled PDE–ODE model (1). The dimensions of the variables and parameters of the model are defined as follows:

(2)

where [z] represents the dimension of z. We assume that the population patches are circular with common radius \(\rho \), which is small compared to the length-scale L of the domain \(\Omega \), and introduce a small scaling parameter \(\varepsilon = \rho /L \ll 1\). The dimensionless variables are defined as follows

$$\begin{aligned} \begin{aligned} P = \frac{L^2}{p_c} {\mathcal {P}}, \quad S_j = \frac{{\mathcal {S}}_j}{{\mathcal {N}}_j}, \quad I_j = \frac{{\mathcal {I}}_j}{{\mathcal {N}}_j},\quad R_j = \frac{{\mathcal {R}}_j}{{\mathcal {N}}_j}, \quad \pmb {x} = \frac{\pmb {X}}{L}, \quad t = \delta \, T. \end{aligned} \end{aligned}$$
(3)

so that \(S_j\), \(I_j\), and \(R_j\) are the proportion of susceptible, infected, and recovered/removed individuals in the \(j{\text {th}}\) patch, respectively, and \(P \equiv P(\pmb {x},t)\) is the dimensionless density of pathogens in the bulk region. Upon substituting (3) into (1), we derive that \(P(\pmb {x},t)\) satisfies

$$\begin{aligned} \frac{\partial P}{\partial t} =&\, D \, \Delta \, P - \, P, \quad t > 0,\quad \pmb {x} \in \Omega \setminus \cup _{j=1}^{m}\,\Omega _{\varepsilon j}; \end{aligned}$$
(4a)
$$\begin{aligned} \, \partial _{n_{\pmb {x}}} \, P = \, 0, \quad \pmb {x} \in \partial \Omega ;&\qquad D \, \partial _{n_{\pmb {x}}} \, P = - r_j \left( \frac{{\mathcal {N}}_{j} L }{\delta \,p_c}\right) I_j, \quad \pmb {x} \in \partial \Omega _{\varepsilon j},\quad j = 1, \dots , m, \end{aligned}$$
(4b)

where \(D \equiv D_B/(\delta \, L^2)\) is the effective diffusion rate of pathogens in the bulk region. From the ODE system ((1c)–(1e)), we derive the dimensionless system of ODEs for the population dynamics of the \(j{\text {th}}\) patch as

$$\begin{aligned} \begin{aligned} \frac{\text {d} S_j}{\text {d}t}&= - \sum _{i =1}^{m} \left( \frac{\mu _{i} }{\delta \,L}\right) S_j\, \gamma _{ji} \int _{\partial \Omega _{\varepsilon i}}\, P \,\,\,\text {d}s_{\pmb {x}}\,;\\ \frac{\text {d} I_j}{\text {d}t}&= \sum _{i =1}^{m} \left( \frac{\mu _{i} }{\delta \,L}\right) S_j \,\gamma _{ji} \int _{\partial \Omega _{\varepsilon i}}\, P \,\,\,\text {d}s_{\pmb {x}} - \phi _j I_j \,; \\ \frac{\text {d} R_j}{\text {d}t}&= \phi _j I_j,\qquad j = 1, \dots , m, \end{aligned} \end{aligned}$$
(5)

where \(\phi _j = \alpha _j/\delta \) is the dimensionless recovery rate and \(\Omega _{\varepsilon j} = \{ \pmb {x} : |\pmb {x}_j - \pmb {x}| < \varepsilon \}\) represents the \(j{\text {th}}\) patch of radius \(\varepsilon \ll 1\) with center at \(\pmb {x}_j\). Note that we have used the scaling \(\text {d}S_{\pmb {X}} = L \,\text {d}s_{\pmb {x}}\) in the integrals on the boundary of the patches. We set

$$\begin{aligned} \frac{ \beta _{j}}{2\pi \varepsilon } = \frac{\mu _j}{\delta L} \qquad \text {and} \qquad \frac{\sigma _{j}}{2 \pi \varepsilon } = r_j\frac{{\mathcal {N}}_j L}{ \delta \, p_c}, \end{aligned}$$
(6)

where \(\beta _j\) and \(\sigma _j\) are \({\mathcal {O}}(1)\). We have assumed that \(\left( \mu _{j} /\delta \,L \right) \) and \(r_j \left( {\mathcal {N}}_{j} L/ \delta \,p_c \right) \) are \( {\mathcal {O}}(1/\varepsilon )\) in order to effectively capture the density of pathogens shed into the bulk region by infected individuals in each patch, since the patches are relatively small compared to the length-scale of the domain. This re-scaling enables us to write the dimensionless transmission and shedding rates, \(\beta _j\) and \(\sigma _j\), respectively, as functions of the circumference of the \(j{\text {th}}\) patch. Upon substituting (6) into (4) and (5), we obtain that the dimensionless density of the pathogens \(P(\pmb {x},t)\) satisfies

$$\begin{aligned} \frac{\partial P}{\partial t} =&\, D \, \Delta \, P - \, P, \quad t > 0,\,\quad \pmb {x} \in \Omega \setminus \cup _{j=1}^{m}\,\Omega _{\varepsilon j}; \end{aligned}$$
(7a)
$$\begin{aligned} \, \partial _{n_{\pmb {x}}} \, P = \, 0, \quad \pmb {x} \in \partial \Omega ;&\qquad 2\pi \varepsilon D \, \partial _{n_{\pmb {x}}} \, P = - \sigma _j \, I_j, \quad \pmb {x} \in \partial \Omega _{\varepsilon j},\qquad j = 1, \dots , m, \end{aligned}$$
(7b)

which is coupled to the dimensionless SIR dynamics of the \(j{\text {th}}\) patch as follows

$$\begin{aligned} \begin{aligned} \frac{\text {d} S_j}{\text {d}t}&= - \frac{1}{2\pi \varepsilon } \sum _{i =1}^{m} \beta _i \, S_j \, \gamma _{ji} \int _{\partial \Omega _{\varepsilon j}} P \,\,\,\text {d}s_{\pmb {x}}; \\ \frac{\text {d} I_j}{\text {d}t}&= \frac{1}{2\pi \varepsilon } \sum _{i =1}^{m} \beta _i \, S_j \, \gamma _{ji} \int _{\partial \Omega _{\varepsilon j}} P \,\,\,\text {d}s_{\pmb {x}} - \phi _j I_j; \\ \frac{\text {d} R_j}{\text {d}t}&= \phi _j I_j,\qquad \qquad j = 1, \dots , m, \end{aligned} \end{aligned}$$
(7c)

where

$$\begin{aligned} \beta _{j} = \frac{2\pi \varepsilon \, }{\delta L}\mu _j, \qquad \sigma _{j} = \frac{2 \pi \varepsilon }{ \delta \, p_c}r_j\,{\mathcal {N}}_j L \qquad \text {and} \qquad \phi _j = \frac{\alpha _j}{\delta }. \end{aligned}$$
(8)

are the dimensionless transmission, shedding, and recovery rates for the \(j{\text {th}}\) patch, respectively. Table 1 shows all parameters and their respective descriptions.

Next, we study the dimensionless coupled model (7) in the limit of fast diffusing pathogens, where \(D = {\mathcal {O}}(\nu ^{-1}) \gg 1\), with \(\nu = -1/\log _e(\varepsilon )\) and \(\varepsilon \ll 1\), using the method of matched asymptotic expansions. We define

$$\begin{aligned} D = \frac{D_0}{\nu }, \qquad \text {where} \quad D_0 = {\mathcal {O}}(1) \quad \text {and} \quad \nu = - \frac{1}{\log _e(\varepsilon )} \ll 1. \end{aligned}$$
(9)

Substituting \(D = D_0/\nu \) into (7a) and (7b), we obtain

$$\begin{aligned} \frac{\partial P}{\partial t} =&\, \frac{D_0}{\nu } \, \Delta \, P - \, P, \quad t > 0,\quad \pmb {x} \in \Omega \setminus \cup _{j=1}^{m}\,\Omega _{\varepsilon j}; \end{aligned}$$
(10a)
$$\begin{aligned} \partial _{n_{\pmb {x}}} \, P = \, 0, \quad \pmb {x} \in \partial \Omega ;&\qquad 2\pi \varepsilon \frac{D_0}{\nu } \, \partial _{n_{\pmb {x}}} \, P = - \sigma _j \, I_j, \quad \pmb {x} \in \partial \Omega _{\varepsilon j},\qquad j = 1, \dots , m, \end{aligned}$$
(10b)

Following the approach of David et al. (2020), we derive a two-term asymptotic expansion in terms of \(\nu \) for the density of pathogens in the bulk region given by

$$\begin{aligned} \begin{aligned} P = P_0 + \frac{\nu }{D_0} \sum _{i=1}^{m} \, \sigma _i I_i\, G(\pmb {x};\pmb {x}_i) + \dots , \end{aligned} \end{aligned}$$
(11)

where \(P_0 \equiv P_0(t)\) is the average density of pathogens in the bulk region which satisfies

$$\begin{aligned} \frac{\text {d} P_{0}}{\text {d}t} = - P_0 + \frac{1}{|\Omega |} \sum _{j=1}^{m} \sigma _j \, I_j, \end{aligned}$$
(12)

and \(G(\pmb {x};\pmb {x}_j) \) is the Neumann’s Green function satisfying

$$\begin{aligned} \Delta G&= \frac{1}{|\Omega |} - \delta (\pmb {x} - \pmb {x}_j), \quad \pmb {x} \in \Omega ;\qquad \partial _n G = 0, \quad \pmb {x} \in \partial \Omega ; \end{aligned}$$
(13a)
$$\begin{aligned} G(\pmb {x};\pmb {x}_j) \sim -&\frac{1}{2\pi } \log _e|\pmb {x} - \pmb {x}_j| + {\mathfrak {R}}_j, \quad \text {as} \quad \pmb {x} \rightarrow \pmb {x}_j, \quad \text {and} \quad \int _{\Omega } \, G \, \text {d}\pmb {x} =0, \end{aligned}$$
(13b)

with regular part \({\mathfrak {R}}_j \equiv {\mathfrak {R}}(\pmb {x}_j)\). Similarly, near the \(j{\text {th}}\) population patch, we derive

$$\begin{aligned} \begin{aligned} Q_j&= \left( P_0(t) + \frac{\sigma _j \, I_j }{ 2 \pi D_0} \right) + \frac{\nu }{D_0} \left[ - \left( \frac{\sigma _j \, I_j }{ 2 \pi } \right) \, \log _e \left( \frac{|\pmb {x} - \pmb {x}_j|}{\varepsilon } \right) \right. \\&\qquad \left. + \sigma _j I_j {\mathfrak {R}}_j + \sum _{i \ne j}^{m} \, \sigma _i I_i\, G(\pmb {x}_j;\pmb {x}_i) \right] + \cdots ,\qquad j = 1, \dots , m, \end{aligned} \end{aligned}$$
(14)

On the boundary of the \(j{\text {th}}\) patch, where \(|\pmb {x} - \pmb {x}_j| = \varepsilon \), (14) reduces to

$$\begin{aligned} Q_j= & {} \left( P_0(t) + \frac{\sigma _j \, I_j }{ 2 \pi D_0} \right) \nonumber \\&+ \frac{\nu }{D_0} \left( \sigma _j I_j {\mathfrak {R}}_j + \sum _{i \ne j}^{m} \, \sigma _i I_i\, G(\pmb {x}_j;\pmb {x}_i) \right) + \dots ,\qquad j = 1, \dots , m, \end{aligned}$$
(15)

To couple the dynamics of the diffusing pathogens to that of human population in the \(j{\text {th}}\) patch, we substitute (15) into the ODE system (7c) to obtain

$$\begin{aligned} \begin{aligned} \frac{\text {d} S_j}{\text {d}t}&= - \sum _{i =1}^{m} \beta _i S_j \gamma _{ji} \left( p(t) + \frac{\sigma _i \, I_i }{ 2 \pi D_0} \right) - \frac{\nu }{D_0} \sum _{i =1}^{m} \beta _i S_j \gamma _{ji} \, \Psi _i \\ \frac{\text {d} I_j}{\text {d}t}&= \sum _{i =1}^{m} \beta _i S_j \gamma _{ji} \left( p(t) + \frac{\sigma _i \, I_i }{ 2 \pi D_0} \right) - \frac{\nu }{D_0} \sum _{i =1}^{m} \beta _i S_j \gamma _{ji} \, \Psi _i - \phi _j I_j \\ \frac{\text {d} R_j}{\text {d}t}&= \phi _j I_j,\qquad j = 1, \dots , m, \end{aligned} \end{aligned}$$
(16)

where \(\Psi _j = ( {\mathcal {G}}\Phi )_j \) is the \(j{\text {th}}\) entry of the vector \({\mathcal {G}}\Phi \), with \(\Phi = (\sigma _1 I_1, \dots , \sigma _m I_m)^T\). Here, \({\mathcal {G}}\) is the Neumann–Green’s matrix whose entries are defined by

$$\begin{aligned} ({\mathcal {G}})_{jj} = {\mathfrak {R}}(\pmb {x}_j) \quad \text {for} \quad i = j \quad \text {and} \quad ({\mathcal {G}})_{ij} = ({\mathcal {G}})_{ji} = G(\pmb {x}_i;\pmb {x}_j) \quad \text {for} \quad i \ne j,\qquad \end{aligned}$$
(17)

where \(G(\pmb {x}_j;\pmb {x}_i)\) is the Neumann–Green’s function satisfying (13) and \({\mathfrak {R}}_j \equiv {\mathfrak {R}}(\pmb {x}_j) \) is its regular part at the point \(\pmb {x} = \pmb {x}_j\). For convenience of notation, we have replaced \(P_0(t)\) with p(t) in (16). Combining (12) and (16), we obtain an ODE systems for the average density of pathogen in the atmosphere coupled to the population dynamics in the patches that is valid in the limit \(D = {\mathcal {O}}(\nu ^{-1})\), where \(\nu = -1/\log (\varepsilon )\) with \(\varepsilon \ll 1\). This ODE system is given by

$$\begin{aligned} \frac{\text {d} p(t)}{\text {d}t}&= - p(t) + \frac{1}{|\Omega |} \sum _{j=1}^{m} \sigma _j \, I_j , \end{aligned}$$
(18a)
$$\begin{aligned} \frac{\text {d} S_j}{\text {d}t}&= - \sum _{i =1}^{m} \beta _i S_j \gamma _{ji} \left( p(t) + \frac{\sigma _i \, I_i }{ 2 \pi D_0} \right) - \frac{\nu }{D_0} \sum _{i =1}^{m} \beta _i S_j \gamma _{ji} \, \Psi _i , \end{aligned}$$
(18b)
$$\begin{aligned} \frac{\text {d} I_j}{\text {d}t}&= \sum _{i =1}^{m} \beta _i S_j \gamma _{ji} \left( p(t) + \frac{\sigma _i \, I_i }{ 2 \pi D_0} \right) - \frac{\nu }{D_0} \sum _{i =1}^{m} \beta _i S_j \gamma _{ji} \, \Psi _i - \phi _j I_j, \end{aligned}$$
(18c)
$$\begin{aligned} \frac{\text {d} R_j}{\text {d}t}&= \phi _j I_j,\qquad \qquad j = 1, \dots , m. \end{aligned}$$
(18d)

Note that the terms with \(\sigma _j \, I_j /(2 \pi D_0) \) in (18) do not model direct transmission, but rather they account for the pathogens shed by infected individuals in the \(j{\text {th}}\) patch. The density of these pathogens depends on the scaled-diffusion rate \(D_0\) and the population of infected individuals in the patch. When the pathogens diffuse slowly (small \(D_0\)), there is significant contribution from this term, and this contribution decreases as \(D_0\) increases. In the limit \(D_0 \rightarrow \infty \), these terms go to zero and the ODE system reduces to the model for well-mixed regime present in equation 5 of Funke (2018). In Sects. 3, we study the reduced ODE system (18) for two population patches and compute the basic reproduction number and final size relation for this scenario.

3 Two-Patch Model with Heterogeneous Mixing

In this section, we consider a scenario with two population patches (\(m=2\)) centered at \(\pmb {x}_1 = (0.5, 0)\) and \(\pmb {x}_2 = (-0.5, 0)\) in the unit disk and use the dimensionless coupled PDE–ODE model (7) and the reduced system of ODEs (18) to study the dynamics of the disease in these populations. For this scenario, we derive from (7) that the dimensionless density of pathogens \(P(\pmb {x},t)\) in the atmosphere satisfies

$$\begin{aligned} \frac{\partial P}{\partial t}&= \, D \, \Delta \, P - \, P, \quad t > 0,\quad \pmb {x} \in \Omega \setminus \{ \Omega _1 \cup \Omega _2 \}; \end{aligned}$$
(19a)
$$\begin{aligned} \, \partial _{n} \, P&= \, 0, \quad \pmb {x} \in \partial \Omega ; \qquad 2\pi \varepsilon D \, \partial _{n} \, P = - \sigma _1 \, I_1, \quad \pmb {x} \in \partial \Omega _1; \nonumber \\ 2\pi \varepsilon D \, \partial _{n} \, P&= - \sigma _2 \, I_2, \quad \pmb {x} \in \partial \Omega _2, \end{aligned}$$
(19b)

where \(\Omega _1\) and \(\Omega _2\) are the two population patches centered at \(\pmb {x}_1 = (0.5, 0)\) and \(\pmb {x}_2 = (-0.5, 0)\), respectively. The dynamics of the diffusion pathogens is coupled to the population dynamics of the two patches as follows

$$\begin{aligned}&\mathbf{Patch} 1&\mathbf{Patch} 2 \nonumber \\ \frac{\text {d} S_1}{\text {d}t}&= - \frac{1}{2\pi \varepsilon } \sum _{i =1}^{2} \beta _i \, S_1 \, \gamma _{1i} \int _{\partial \Omega _{\varepsilon _i}}\, P \,\,\,\text {d}s_{\pmb {x}} ;&\frac{\text {d} S_2}{\text {d}t}&= - \frac{1}{2\pi \varepsilon } \sum _{i =1}^{2} \beta _i \, S_2 \, \gamma _{2i} \int _{\partial \Omega _{\varepsilon _i}}\, P \,\text {d}s_{\pmb {x}} ; \nonumber \\ \frac{\text {d} I_1}{\text {d}t}&= \frac{1}{2\pi \varepsilon } \sum _{i =1}^{2} \beta _i \, S_1 \, \gamma _{1i} \int _{\partial \Omega _{\varepsilon _i}}\, P \,\,\,\text {d}s_{\pmb {x}} - \phi _1 I_1;&\frac{\text {d} I_2}{\text {d}t}&= \frac{1}{2\pi \varepsilon } \sum _{i =1}^{2} \beta _i \, S_2 \, \gamma _{2i} \int _{\partial \Omega _{\varepsilon _i}}\, P \,\text {d}s_{\pmb {x}} - \phi _2 I_2; \nonumber \\ \frac{\text {d} R_1}{\text {d}t}&= \phi _1 I_1;&\frac{\text {d} R_2}{\text {d}t}&= \phi _2 I_2. \end{aligned}$$
(19c)

Similarly, we derive the reduced ODE system for the two population patch scenario from (18) as

$$\begin{aligned} \frac{\text {d} p}{\text {d}t}&= - p + \frac{1}{|\Omega |} ( \sigma _1 \, I_1 + \sigma _2 \, I_2),\qquad \qquad \qquad \qquad \qquad \end{aligned}$$
(20a)
$$\begin{aligned}&\mathbf{Patch} 1&\mathbf{Patch} 2 \nonumber \\ \frac{\text {d} S_1}{\text {d}t}&= - \sum _{i =1}^{2} \beta _i \, S_1 \, \gamma _{1i} \left( p(t) + \frac{\sigma _i \, I_i }{ 2 \pi D_0} \right) ,&\frac{\text {d} S_2}{\text {d}t}&= - \sum _{i =1}^{2} \beta _i \, S_2 \, \gamma _{2i} \left( p(t) + \frac{\sigma _i \, I_i }{ 2 \pi D_0} \right) ,\nonumber \\ \frac{\text {d} I_1}{\text {d}t}&= \sum _{i =1}^{2} \beta _i \, S_1 \, \gamma _{1i} \left( p(t) + \frac{\sigma _i \, I_i }{ 2 \pi D_0} \right) - \phi _1 I_1,&\frac{\text {d} I_2}{\text {d}t}&= \sum _{i =1}^{2} \beta _i \, S_2 \, \gamma _{2i} \left( p(t) + \frac{\sigma _i \, I_i }{ 2 \pi D_0} \right) - \phi _2 I_2, \nonumber \\ \frac{\text {d} R_1}{\text {d}t}&= \phi _1 I_1,&\frac{\text {d} R_2}{\text {d}t}&= \phi _2 I_2, \end{aligned}$$
(20b)

where \(p \equiv p(t)\) is the average density of pathogens in the atmosphere and \(D_0\) is the scaled diffusion coefficient of the pathogens. The ODE system (20) has a similar structure to those studied in David et al. (2020) and Funke (2018). The coupled PDE–ODE model (19), and the reduced ODE system (20) will be used to study the transmission dynamics of the disease for two populations scenario.

3.1 Basic Reproduction Number

Here, we use the reduced system of ODEs (20) to compute the basic reproduction number, \({\mathcal {R}}_0\) (the secondary infections caused by a single infective into a totally susceptible population). To use the next generational matrix approach for computing \({\mathcal {R}}_0\) as was done in Odo et al. (1990), Holland (2007) and Van den Driessche and Watmough (2002), we construct a system of equations for the infectious classes given by

$$\begin{aligned} \begin{aligned} \frac{\text {d} I_1}{\text {d}t}&= \beta _1 S_1 \gamma _{11} \left( p(t) + \frac{\sigma _1 \, I_1 }{ 2 \pi D_0} \right) + \beta _2 S_1 \gamma _{12} \left( p(t) + \frac{\sigma _2 \, I_2 }{ 2 \pi D_0} \right) - \phi _1 I_1,\\ \frac{\text {d} I_2}{\text {d}t}&= \beta _1 S_2 \gamma _{21} \left( p(t) + \frac{\sigma _1 \, I_1 }{ 2 \pi D_0} \right) + \beta _2 S_2 \gamma _{22} \left( p(t) + \frac{\sigma _2 \, I_2 }{ 2 \pi D_0} \right) - \phi _2 I_2,\\ \frac{\text {d} p}{\text {d}t}&= - p + \frac{1}{|\Omega |} ( \sigma _1 \, I_1 + \sigma _2 \, I_2), \end{aligned} \end{aligned}$$
(21)

where \(\gamma _{11}\) and \(\gamma _{12}\) are the proportions of the residence of patch 1 that are currently in patch 1 and patch 2, respectively, with \(\gamma _{11} + \gamma _{12} = 1\). Similarly, \(\gamma _{21}\) and \(\gamma _{22}\) are the proportions of the residence of patch 2 that are currently in patch 1 and patch 2, respectively, with \(\gamma _{21} + \gamma _{22} = 1\).

At the disease-free equilibrium DFE \(=(S_{1}^{*}, I_{1}^{*}, R_{1}^{*}, S_{2}^{*}, I_{2}^{*}, R_{2}^{*}, p^{*}) \equiv (S_1(0), 0, 0, S_2(0), 0, 0, 0)\), we construct the Jacobian matrix F for new infections and the matrix V for transfer of infections. Since \(I_1(0) = I_2(0) = R_1(0) = R_2(0) = 0\), at the DFE, we have \(S_1^{*} = N_1(0)\) and \(S_2^{*} = N_2(0)\). Based on this, the matrix F is given by

$$\begin{aligned} F=\left( \frac{\partial {\mathcal {F}}_i}{\partial x_j} \right) _{i,j}=\begin{pmatrix} \displaystyle \frac{\sigma _1 \beta _1 \gamma _{11}N_1(0)}{2 \pi D_0} &{}\displaystyle \frac{\sigma _2 \beta _2 \gamma _{12} N_1(0)}{2 \pi D_0} &{}\displaystyle \left( \beta _1 \gamma _{11} + \beta _2 \gamma _{12} \right) N_1(0)\\ \\ \displaystyle \frac{\sigma _1 \beta _1 \gamma _{21}N_2(0)}{2 \pi D_0} &{}\displaystyle \frac{\sigma _2 \beta _2 \gamma _{22} N_2(0)}{2 \pi D_0} &{}\displaystyle \left( \beta _1 \gamma _{21} + \beta _2 \gamma _{22} \right) N_2(0)\\ \\ 0 &{} 0 &{} 0 \end{pmatrix},\qquad \end{aligned}$$
(22)

where the functions \(({\mathcal {F}}_1, {\mathcal {F}}_2, {\mathcal {F}}_2) \equiv (I'_1, I'_2, p')\) are as given in (21) with \((x_1, x_2, x_3) \equiv (I_1, I_2,p)\). Similarly, from (21), we construct the Jacobian matrix V for the rates of transfer of individuals between the infected compartments as

$$\begin{aligned} V=\left( \frac{\partial {\mathcal {V}}_i}{\partial x_j} \right) _{i,j}=\begin{pmatrix} \displaystyle \phi _1 &{} 0 &{} 0 \\ \\ 0 &{} \displaystyle \phi _2 &{} 0 \\ \\ - \displaystyle \frac{\sigma _1}{| \Omega |} &{} - \displaystyle \frac{\sigma _2}{| \Omega |} &{} 1 \end{pmatrix}. \end{aligned}$$
(23)

Upon finding the inverse of V and multiplying by F (22) from the left, we obtain the next generation matrix defined by \({\mathcal {M}} = FV^{-1}\). Our desired basic reproduction number is the spectral radius of the next generation matrix. Computing the eigenvalues of this matrix, we obtain the basic reproduction number of our model as

$$\begin{aligned} {\mathcal {R}}_0 = \displaystyle \frac{\mho _{11} + \mho _{22} }{4 \pi \phi _1 \phi _2 |\Omega | D_0} + \displaystyle \frac{\sqrt{\mho ^2_{11} + \Upsilon + \mho ^2_{22}}}{2 \phi _1 \phi _2}, \end{aligned}$$
(24)

where \(\mho _{11}\), \(\mho _{22}\), and \(\Upsilon \) are defined as follows

$$\begin{aligned} \mho _{11}= & {} \displaystyle \left( 2 \, \pi \, D_0 (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) + \beta _1 \, \gamma _{11} \, |\Omega | \right) N_1(0) \, \phi _2 \, \sigma _1,\nonumber \\ \mho _{22}= & {} \displaystyle \left( 2 \, \pi \, D_0 (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) + \beta _2 \, \gamma _{22} \, |\Omega | \right) N_2(0) \, \phi _1 \, \sigma _2, \nonumber \\ \Upsilon= & {} 2 \, \phi _1 \, \phi _2 \, \sigma _1 \, \sigma _2 \, N_1(0) \, N_2(0) \left( \frac{(\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12})}{2 \, \pi \, D_0 \, |\Omega |^2} \mho ^-_{22} - \frac{\beta _1 \, \gamma _{11}}{4 \, \pi ^2 \, D^2_0 \, |\Omega |} \mho _{22} \right. \nonumber \\&\left. + \frac{\beta _1 \, \gamma _{21}}{4 \, \pi ^2 \, D^2_0 \, |\Omega |} \mho ^-_{12} + \frac{\beta _2 \, \gamma _{12}}{4 \, \pi ^2 \, D^2_0 \, |\Omega |} \mho ^-_{21} \right) . \end{aligned}$$
(25)

Here, the variables \(\mho ^-_{22}\), \(\mho ^-_{12}\) and \(\mho ^-_{21}\) are given by

$$\begin{aligned} \begin{aligned}&\mho ^-_{22} = \Big ( 2 \, \pi \, D_0 (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) - \beta _2 \, \gamma _{22} \, |\Omega | \Big ), \\&\mho ^-_{12} = \Big ( 4 \, \pi \, D_0 (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) + \beta _2 \, \gamma _{12} \, |\Omega | \Big ), \\&\mho ^-_{21} = \Big ( 4 \, \pi \, D_0 (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) + \beta _1 \, \gamma _{21} \, |\Omega | \Big ), \end{aligned} \end{aligned}$$
(26)

To better understand the basic reproduction number \({\mathcal {R}}_0\) (24), we consider some limiting scenarios and make simplifying assumptions on the mixing pattern between the two populations:

  1. 1.

    In the well-mixed limit, where \(D_0 \rightarrow \infty \), the basic reproduction number in (24) reduces to

    $$\begin{aligned} {\mathcal {R}}_0^{\infty }= (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_1 + (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_2, \end{aligned}$$
    (27)

    where \({\mathcal {R}}_1 = N_1(0)\, \sigma _1/ (\phi _1 |\Omega |)\) and \({\mathcal {R}}_2 = N_2(0) \, \sigma _2 / (\phi _2 |\Omega |)\). The first term in (27) given by \((\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_1\) accounts for the secondary infections caused indirectly by a quantity \(\sigma _1\) of the pathogen shed by a single infected individual in patch 1 per unit time for a time period \(1/ \phi _1\). A similar explanation holds for the second term in (27) given by \((\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_2\), but in terms of patch 2.

  2. 2.

    For a symmetric proportionate mixing pattern, where \(\gamma _{11} = \gamma _{21} =\gamma _1\), and \(\gamma _{12} = \gamma _{22} =\gamma _2\), with \(\gamma _1 + \gamma _2 =1\) and \(\gamma _{11} \, \gamma _{22} - \gamma _{12} \, \gamma _{21}=0\), the reproduction number in (27) becomes

    $$\begin{aligned} {\mathcal {R}}_0^{\infty } = \displaystyle \frac{(\beta _1 \, \gamma _1 + \beta _2 \, \gamma _2) N_1(0) \sigma _1 }{\phi _1 |\Omega |} + \displaystyle \frac{(\beta _1 \, \gamma _1 + \beta _2 \, \gamma _2) N_2(0) \sigma _2 }{\phi _2 |\Omega |}. \end{aligned}$$
    (28)
  3. 3.

    When there is no mobility, that is, the members of each patch only mix with individuals in their patch, we have \(\gamma _{11} = \gamma _{22} =1\) (consequently, \(\gamma _{12} = \gamma _{21} =0\)). Therefore, the reproduction number in (27) reduces to

    $$\begin{aligned} {\mathcal {R}}_0^{\infty } = \displaystyle \frac{\beta _1 N_1(0) \sigma _1 }{\phi _1 |\Omega |} + \displaystyle \frac{\beta _2 N_2(0) \sigma _2 }{\phi _2 |\Omega |}. \end{aligned}$$
    (29)

    Note that this scenario is the same as that of section 4 of David et al. (2020), and the reduced reproduction number (29) for this scenario is the same as equation 4.9 of David et al. (2020).

Our numerical simulations will give further explanations of \({\mathcal {R}}_0\) (for the case where \(D_0={{\mathcal {O}}}(1)\)) and \({\mathcal {R}}_0^{\infty }\) (for the well-mixed limit \(D_0\rightarrow \infty \)). We summarize the implications of the reproduction number \({\mathcal {R}}_0^{\infty }\) in the following easily proved theorem.

Theorem 1

For system (20), the infection dies out whenever \({\mathcal {R}}_0^{\infty }<1\), while the disease remains stable in the community but not cause an epidemic whenever \({\mathcal {R}}_0^{\infty }=1\). Contrarily, an epidemic occurs whenever \({\mathcal {R}}_0^{\infty }>1\).

3.2 Final Size Relation

To represent the epidemic size in terms of the basic reproduction number and the model parameters, we derive a final size relation for the two-patch epidemic model (18). Following the approach used in Arino and Brauer (2007), Brauer (2008a), Brauer (2008b), Brauer (2017a), Brauer (2017b), Brauer (2019), Brauer and Castillo-Chaavez (2012), Brauer et al. (2018), David et al. (2020) and Funke (2018), we obtain

$$\begin{aligned} \begin{aligned} \log \frac{S_{1,0}}{S_{1,\infty }}&= (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, p_0 + (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_1 \left\{ 1-\frac{S_{1, \infty }}{N_1(0)} \right\} \\&\quad + \frac{ \sigma _1 \, N_1(0) \, \beta _1 \, \gamma _{11}}{2 \, \pi \, D_0 \, \phi _1} \left\{ 1-\frac{S_{1 , \infty }}{N_1(0)} \right\} \\&+ (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_2 \left\{ 1-\frac{S_{2 , \infty }}{N_2(0)} \right\} + \frac{ \sigma _2 \, N_2(0) \, \beta _2 \, \gamma _{12}}{2 \, \pi \, D_0 \, \phi _2} \left\{ 1-\frac{S_{2 , \infty }}{N_2(0)} \right\} , \end{aligned} \end{aligned}$$
(30a)

and

$$\begin{aligned} \begin{aligned} \log \frac{S_{2,0}}{S_{2 ,\infty }} =&(\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, p_0 + (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_1 \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} \\&+ \frac{ \sigma _1 \, N_1(0) \, \beta _1 \, \gamma _{21}}{2 \, \pi \, D_0 \, \phi _1} \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} \\&+ (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_2 \left\{ 1-\frac{S_{2,\infty }}{N_2(0)} \right\} + \frac{ \sigma _2 \, N_2(0) \, \beta _2 \, \gamma _{22}}{2 \, \pi \, D_0 \, \phi _2} \left\{ 1-\frac{S_{2,\infty }}{N_2(0)} \right\} , \end{aligned} \end{aligned}$$
(30b)

where \({\mathcal {R}}_1\) and \({\mathcal {R}}_2\) are as defined in (27), \(p_0\) is the initial average density of pathogens, \(N_1(0)\) and \(N_2(0)\) are the initial populations in patch 1 and patch 2, respectively. Here, \(S_{1,0}\) and \(S_{2, 0}\) denote the initial susceptible populations in the patches 1 and 2, respectively, while \(S_{1,\infty }\) and \(S_{2, \infty }\) are the susceptible populations left in patches 1 and 2 after the outbreak. The remaining parameters are defined in Table 1.

The final size relations in (30a) and (30b) for patches 1 and 2, respectively, imply that \(S_{1, \infty } > 0\) and \(S_{2, \infty } > 0\). They give the relationship between the basic reproduction number \(R_0\) and the final epidemic size in patches 1 and 2, respectively. Note that the total number of infected individuals in patches 1 and 2 over the epidemic period is, respectively, given by \(N_1(0) - S_{1, \infty }\) and \(N_2(0) - S_{2, \infty }\), which can be described in terms of the attack rates/ratios as \(\displaystyle \left[ 1- \frac{S_{1, \infty }}{N_1(0)} \right] \) and \(\displaystyle \left[ 1- \frac{S_{2, \infty }}{N_2(0)} \right] \) as in Brauer (2008b). The final size relation (30) takes a simpler form using the following assumptions.

  1. 1.

    In the case where the outbreak begins with infected individuals and no pathogens, that is, \(I_1(0) \ne 0\), \(I_2(0) \ne 0\) and \(p_0 =0\), the final size relation for patches 1 and 2 in (30a) and (30b) can be written as

    $$\begin{aligned} \log \frac{S_{1,0}}{S_{1,\infty }}= & {} (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_1 \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} + \frac{ \sigma _1 \, N_1(0) \, \beta _1 \, \gamma _{11}}{2 \, \pi \, D_0 \, \phi _1} \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} \nonumber \\&+ (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_2 \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} + \frac{ \sigma _2 \, N_2(0) \, \beta _2 \, \gamma _{12}}{2 \, \pi \, D_0 \, \phi _2} \left\{ 1-\frac{S_{2,\infty }}{N_2(0)} \right\} ,\nonumber \\ \end{aligned}$$
    (31)

    and

    $$\begin{aligned} \log \frac{S_{2,0}}{S_{2,\infty }}= & {} (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_1 \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} + \frac{ \sigma _1 \, N_1(0) \, \beta _1 \, \gamma _{21}}{2 \, \pi \, D_0 \, \phi _1} \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} \nonumber \\&+ (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_2 \left\{ 1-\frac{S_{2,\infty }}{N_2(0)} \right\} + \frac{ \sigma _2 \, N_2(0) \, \beta _2 \, \gamma _{22}}{2 \, \pi \, D_0 \, \phi _2} \left\{ 1-\frac{S_{2,\infty }}{N_2(0)} \right\} .\nonumber \\ \end{aligned}$$
    (32)
  2. 2.

    In the limit \(D_0 \rightarrow \infty \) (well mixed), the final size relation (31) and (32) becomes

    $$\begin{aligned} \begin{aligned} \log \frac{S_{1,0}}{S_{1,\infty }}&= (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_1 \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} + (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_2 \left\{ 1-\frac{S_{2,\infty }}{N_2(0)} \right\} ,\\ \log \frac{S_{2,0}}{S_{2,\infty }}&= (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_1 \left\{ 1-\frac{S_{1,\infty }}{N_1(0)} \right\} + (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_2 \left\{ 1-\frac{S_{2,\infty }}{N_2(0)} \right\} . \end{aligned}\nonumber \\ \end{aligned}$$
    (33)

    This result can be written in a matrix form as

    $$\begin{aligned} \begin{aligned}&\left( \! \begin{array}{c} \displaystyle \log \frac{S_{1,0}}{S_{1, \infty }} \\ \\ \displaystyle \log \frac{S_{2,0}}{S_{2, \infty }} \end{array} \!\right) = \left( \! \begin{array}{cc} {\mathbb {M}}_{11} &{} {\mathbb {M}}_{12}\\ \\ {\mathbb {M}}_{21} &{} {\mathbb {M}}_{22} \end{array} \!\right) \left( \! \begin{array}{c} \displaystyle 1-\frac{S_{1,\infty }}{N_1 (0)}\\ \\ \displaystyle 1-\frac{S_{2,\infty }}{N_2 (0)} \end{array} \!\right) ,\\&\text{ where } \quad {\mathbb {M}} = \left( \! \begin{array}{cc} \displaystyle (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_1 &{} \displaystyle (\beta _1 \, \gamma _{11} + \beta _2 \, \gamma _{12}) \, {\mathcal {R}}_2 \\ \\ \displaystyle (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_1 &{} \displaystyle (\beta _1 \, \gamma _{21} + \beta _2 \, \gamma _{22}) \, {\mathcal {R}}_2 \end{array} \!\right) . \end{aligned} \end{aligned}$$
    (34)
  3. 3.

    If the mixing is proportionate, that is, \(\gamma _{11} =\gamma _{21} =\gamma _1\), and \(\gamma _{12} =\gamma _{22} =\gamma _2\), the final size relation (34) becomes

    $$\begin{aligned}&\left( \! \begin{array}{c} \displaystyle \log \frac{S_{1,0}}{S_{1, \infty }} \\ \\ \displaystyle \log \frac{S_{2,0}}{S_{2, \infty }} \end{array} \!\right) = \left( \! \begin{array}{cc} {\mathbb {N}}_{11} &{} {\mathbb {N}}_{12}\\ \\ {\mathbb {N}}_{21} &{} {\mathbb {N}}_{22} \end{array} \!\right) \left( \! \begin{array}{c} \displaystyle 1-\frac{S_{1,\infty }}{N_1 (0)}\\ \\ \displaystyle 1-\frac{S_{2,\infty }}{N_2 (0)} \end{array} \!\right) ,\nonumber \\&\text{ where } \quad {\mathbb {N}} = \left( \! \begin{array}{cc} \displaystyle (\beta _1 \, \gamma _1 + \beta _2 \, \gamma _2) \, {\mathcal {R}}_1 &{} \displaystyle (\beta _1 \, \gamma _1 + \beta _2 \, \gamma _2) \, {\mathcal {R}}_2 \\ \\ \displaystyle (\beta _1 \, \gamma _1 + \beta _2 \, \gamma _2) \, {\mathcal {R}}_1 &{} \displaystyle (\beta _1 \, \gamma _1 + \beta _2 \, \gamma _2) \, {\mathcal {R}}_2 \end{array} \!\right) . \end{aligned}$$
    (35)
  4. 4.

    If the mixing is like-with-like (no mobility), that is \(\gamma _{11} =\gamma _{22} =1\), and \(\gamma _{12} =\gamma _{21} =0\), the final size relation (34) reduces to

    $$\begin{aligned} \left( \! \begin{array}{c} \displaystyle \log \frac{S_{1,0}}{S_{1, \infty }} \\ \\ \displaystyle \log \frac{S_{2,0}}{S_{2, \infty }} \end{array} \!\right) = \left( \! \begin{array}{cc} {\mathbb {W}}_{11} &{} {\mathbb {W}}_{12}\\ \\ {\mathbb {W}}_{21} &{} {\mathbb {W}}_{22} \end{array} \!\right) \left( \! \begin{array}{c} \displaystyle 1-\frac{S_{1,\infty }}{N_1 (0)}\\ \\ \displaystyle 1-\frac{S_{2,\infty }}{N_2 (0)} \end{array} \!\right) , \quad \text{ where } \quad {\mathbb {W}} = \left( \! \begin{array}{cc} \displaystyle \beta _1 \, {\mathcal {R}}_1 &{} \displaystyle \beta _1 \, {\mathcal {R}}_2 \\ \\ \displaystyle \beta _2 \, {\mathcal {R}}_1 &{} \displaystyle \beta _2 \, {\mathcal {R}}_2 \end{array} \!\right) ,\quad \end{aligned}$$
    (36)

    which can be written as

    $$\begin{aligned} \beta _2 \, \log \frac{S_{1,0}}{S_{1, \infty }} = \beta _1 \, \log \frac{S_{2,0}}{S_{2, \infty }}. \end{aligned}$$
    (37)

    We can further reduce (37) to obtain

    $$\begin{aligned} \displaystyle \left[ \frac{S_{1,0}}{S_{1, \infty }} \right] ^{\, \displaystyle \beta _2} = \displaystyle \left[ \frac{S_{2,0}}{S_{2, \infty }} \right] ^{\, \displaystyle \beta _1} \end{aligned}$$
    (38)

    If \(\beta _1 > \beta _2\) in (38), then

    $$\begin{aligned} 1- \log \frac{S_{1,0}}{S_{1, \infty }} > 1 - \log \frac{S_{2,0}}{S_{2, \infty }}, \end{aligned}$$
    (39)

    which implies that the attack rate/ratio is greater in the more active patch.

The assumptions above and their effects on the epidemic peak and time will be explored and discussed in the numerical simulations.

Fig. 1
figure 1

Effect of pathogen diffusion and human mobility on disease dynamics (ODE model). Numerical simulations of the ODE system (20) for different diffusion rate of pathogens and human mobility. Left panel: infection starts with no pathogen, with initial conditions \(S_1(0) = 299/300\), \(I_1(0) = 1/300\), \(R_1(0) = 0\), \(S_2(0) = 249/250\), \(I_2(0) = 1/250\), \(R_2(0) = 0\), and \(p(0) = 0\), and right panel: infection starts with only pathogens with initial conditions \(S_1(0) = 300/300, I_1(0) = 0, R_1(0) = 0, S_2(0) = 250/250, I_2(0) = 0, R_2(0) = 0\), and \(p(0) = 1\). Top row: no human mobility (\(\gamma _{11}=1, \gamma _{12}= 0, \gamma _{21}=0\), and \(\gamma _{22}=1\)), middle row: symmetric proportionate mixing (), and bottom row: non-symmetric proportionate mixing (low mobility: and ). All other parameters are given in Table 1. Solid curves are for patch 1 while the dashed curves are for patch 2. Note that the solid and dashed curves are overlapping for the results in the middle row (Color figure online)

Full size image

3.3 Numerical Simulations

We present numerical simulations of the dimensional coupled PDE–ODE model (19) and the reduced system of ODEs (20) for two population patches. For all scenarios considered in this section, these patches are located at \(\pmb {x}_1 = (0.5, 0)\) and \(\pmb {x}_2 = (-0.5, 0)\) for patches 1 and 2, respectively, and the coupled PDE–ODE model is solved using FlexPDE6 PDE solutions Inc (2019). We aim to study the effect of pathogen diffusion and human mobility on the disease dynamics in an heterogeneous mixing environment.

Table 1 Model parameters, descriptions, and values
Full size table

In Fig. 1, we used the reduced ODE system (20) to study the effect of the diffusion rate of pathogens and heterogeneous mixing between two populations on the transmission dynamics of the diseases in the populations. For the results in the left panel (initial conditions: \(S_1(0) = 299/300\), \(I_1(0) = 1/300\), \(R_1(0) = 0\), \(S_2(0) = 249/250\), \(I_2(0) = 1/250\), \(R_2(0) = 0\), and \(p(0) = 0\)), we have only infected individuals with no pathogens at the beginning of the outbreak, while for those in the right panel (initial conditions: \(S_1(0) = 300/300, I_1(0) = 0, R_1(0) = 0, S_2(0) = 250/250, I_2(0) = 0, R_2(0) = 0\), and \(p(0) = 1\)), we have only diffusing pathogens with no infected individuals at the beginning of the epidemic. The second scenario, where the outbreak begins with only pathogens, can be related to a scenario where pathogens diffuse out of an infected population to a completely susceptible population. The solid and dashed curves are for patch 1 and patch 2, respectively. The black, blue, and red curves are for the scaled diffusion rates \(D_0 = 0.128\), \(D_0 = 2.556\), and \(D_0 = 76.687\), respectively. Using the relation for \(D_0\) and D given in (9), these scaled diffusion rates correspond to \(D = 0.5\), \(D = 10\), and \(D = 300\), respectively.

We observe from the results in the top panel of Fig. 1 for the scenario with no human mobility (\(\gamma _{11}=1, \gamma _{12}= 0, \gamma _{21}=0\), and \(\gamma _{22}=1\)) that the epidemic peak decreases and the peak time increases, with increase in the scaled diffusion rate. Overall and in this scenario, patch 2 (dashed curves) experiences a higher peak and shorter peak time for all values of \(D_0\) when compared to patch 1 (solid curves) due to higher transmission and shedding rates in patch 2 (see Table 1 for more details). In addition, the epidemic take-off seems delayed when the outbreak begins with some infectives and no pathogen (left panel), compared to when the outbreak begins with zero infectives and a pathogen (right panel). This observation is more apparent when the diffusion rate is increases (blue and red curves).

Similar results obtained using the full coupled PDE–ODE model (19) are given in the top panel of Fig. 2. In the middle panel of Fig. 1, we have the results for the case of symmetric proportional mixing (\(\gamma _{11} = \gamma _{12}= 0.5\) and \(\gamma _{21}= \gamma _{22}= 0.5\)). For this scenario, the activity level in the two populations is assumed to be the same, and as a result, an equal proportion of the two population are mixing at all time. As observed from the results in the middle panel of Fig. 1, this mixing patterns lead to a well-mixed (homogeneous) system. Even though the transmission and shedding rates of patch 2 are higher than those of patch 1, our numerical simulations predict identical epidemics for the two populations.

Fig. 2
figure 2

Effect of pathogen diffusion and human mobility on disease dynamics (PDE–ODE model). Numerical simulations of the full PDE–ODE model (19) for different diffusion rate of pathogens and human mobility. Left panel: infection starts with no pathogen, with initial conditions \(S_1(0) = 299/300\), \(I_1(0) = 1/300\), \(R_1(0) = 0\), \(S_2(0) = 249/250\), \(I_2(0) = 1/250\), \(R_2(0) = 0\), and \(p(0) = 0\), and right panel: infection starts with only pathogens with initial conditions \(S_1(0) = 300/300, I_1(0) = 0, R_1(0) = 0, S_2(0) = 250/250, I_2(0) = 0, R_2(0) = 0\), and \(p(0) = 1\). Top row: no human mobility (\(\gamma _{11}=1, \gamma _{12}= 0, \gamma _{21}=0\), and \(\gamma _{22}=1\)), middle row: symmetric proportionate mixing (\(\gamma _{11} = \gamma _{12}= 0.5\) and \(\gamma _{21}= \gamma _{22}= 0.5\)), and bottom row: non-symmetric proportionate mixing (low mobility: \(\gamma _{11}=0.8, \gamma _{12}= 0.2\) and \(\gamma _{21}= 0.3, \gamma _{22}=0.7\)). All other parameters are given in Table 1. Solid curves are for patch 1 while the dashed curves are for patch 2. Note that the solid and dashed curves are overlapping for the results in the middle row (Color figure online)

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In the bottom panel of Fig. 1, we consider the case of non-symmetric proportionate mixing (low mobility). For this example, we assume that 80% of the residents of patch 1 mix with only the individuals in their patch, while the remaining 20% mix with residents of patch 2 (\(\gamma _{11}=0.8, \gamma _{12}= 0.2\)). On the other hand, we assume 70% of the residents of patch 2 mix with only members of their patch, while the remaining 30% mix with residents of patch 1 (\(\gamma _{21}= 0.3, \gamma _{22}=0.7\)). We observe from the results for this scenario (bottom panel of Fig. 1) that the epidemic peak is higher in patch 2 (dashed curves) compared to patch 1 (solid curves) due to higher transmission and shedding rates in patch 2. In this scenario, we say that the residence of patch 2 is having more activities than those in patch 1 since a larger fraction of them is mixing with the individuals in patch 1 than those of patch 1 are mixing with them.

Comparing the results in the top (no mobility) and bottom (low mobility) panels of Fig. 1, we notice that there is an increase in the epidemic peaks for patch 1 when there is mobility compared to when there is no mobility (top panel). These differences in the epidemic size between the two scenarios are due to the heterogeneous mixing between the two populations. On the other hand, there is a slight decrease in the epidemic peaks for patch 2 when there is mobility (bottom panel) compared to when there is no mobility (top panel). For all the scenarios considered in Fig. 1, similar results computed using the full PDE model (19) are presented in Fig. 2. Both results agree well.

Fig. 3
figure 3

Effect of population mixing patterns on disease dynamics. Numerical simulations of the ODE system (20) (left panel) and the coupled PDE–ODE model (19) (right panel) for two population patches and different population mixing patterns. The solid and dashed curves are for patch 1 and patch 2, respectively. The black, blue, and red curves are, respectively, for the scenarios with no mobility (\(\gamma _{11}=1, \gamma _{12}= 0, \gamma _{21}=0\), and \(\gamma _{22}=1\)), symmetric mixing (\(\gamma _{11} = \gamma _{12}= 0.5\) and \(\gamma _{21}= \gamma _{22}= 0.5\)) and non-symmetric proportionate mixing (low mobility: \(\gamma _{11}=0.8, \gamma _{12}= 0.2\) and \(\gamma _{21}= 0.3, \gamma _{22}=0.7\)). The initial conditions used are \(S_1(0) = 299/300, I_1(0) = 1/300, R_1(0) = 0, S_2(0) = 249/250, I_2(0) = 1/250, R_2(0) = 0\), and \(p(0) = 1\). For the PDE–ODE model, we used \(D = 10\) (right panel), corresponding to \(D_0 = 2.556\) for the reduce ODE system (left panel). All other parameters are given in Table 1 (Color figure online)

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The results in Fig. 3 are used to study the effect of heterogeneous mixing on the disease dynamics for a fixed diffusion rate of pathogens, \(D = 10\) (corresponding to \(D_0 = 2.556\)). The initial conditions used are \(S_1(0) = 299/300\), \(I_1(0) = 1/300\), \(R_1(0) = 0\), \(S_2(0) = 249/250\), \(I_2(0) = 1/250\), \(R_2(0) = 0\), and \(p(0) = 1\), with other parameters as given in Table 1. Similar to the results presented in Figs. 1 and 2, the solid and dashed curves are for patch 1 and patch 2, respectively. The results in the left panel were obtained using the reduced ODE system (20) and those in right panel were obtained using the coupled PDE–ODE model (19). The black curves show the results for when there is no movement between the two populations, the blue curves are for symmetric proportionate mixing, where half of the two populations are mixing, while the red curves are for non-symmetric proportionate mixing (low mobility). We observe from these results that the no mixing scenario predicts the most difference in the epidemics of the two populations. When there is a symmetric proportionate mixing, an identical epidemic is predicted for the two populations, irrespective of the difference in their transmission and shedding rates. Lastly, for the case of non-symmetric mixing, there is a decrease in the epidemic for the population with higher shedding and transmission rates (patch 1) and an increase in the epidemic for the other population with lower transmission and shedding rates (patch 2). Comparing the results in the left and right panel of Fig. 3, we notice that the results from the reduced system of ODEs (20) agree well with those obtained using the full coupled PDE–ODE model (19). This shows that the reduced ODE system provides a good approximation for the coupled PDE–ODE system.

4 Discussion

We have extended the novel coupled PDE–ODE model developed in David et al. (2020) for studying the transmission dynamics of airborne diseases to include human mobility. Human mobility between populations was incorporated using the Lagrangian approach. This model is used to study the indirect transmission of airborne diseases, where a susceptible individual becomes infected after coming in contact with pathogens. Infected individuals shed pathogens into the environment at some rate, and the pathogens diffuse and decay in the environment at constant rates. In the limit of fast diffusing pathogens, matched asymptotic analysis was used to reduce the PDE–ODE model to a nonlinear system of ODEs for the average density of pathogens in the environment. The reduced ODEs system was then used to derive the basic reproduction number and a final size relation for the epidemic. Numerical simulations of both the coupled PDE–ODE model and the reduced system of ODEs were used to study the effect of the diffusion rate of pathogens and heterogeneous mixing on the dynamics of airborne diseases.

To study the effect of heterogeneous mixing between populations, we considered two non-identical population patches centered at (0.5, 0) and \((-0.5, 0)\) for patches 1 and 2, respectively. The differences between the two patches were introduced through the transmission and shedding rates, where patch 2 has higher transmission and shedding rates relative to patch 1. In addition, we considered three mobility scenarios: no mobility, symmetric proportionate mixing (half populations moving), and non-symmetric proportionate mixing (low mobility). The results of our simulations show that even though the two populations are non-identical, symmetric mixing pattern leads to a well-mixed population with identical epidemics in the two populations, as a result of 50% of members of patch 2 (with higher transmission and shedding rates) mixing with individuals in patch 1 and vice versa. For non-symmetric proportionate mixing, we assumed that 80% of the individuals in patch 1 have contact with only those in patch 1, while the remaining 20% of the population have contacts with those in patch 2. On the other hand, 70% of the population of patch 2 have contact with only those in patch 2, while the remaining 30% have contact with only those in patch 1. For this mobility pattern, our numerical simulations show an increase in the epidemic in patch 1 (the patch with smaller shedding and transmission rates) compared to when there is no movement. We believe that this increase in the epidemic is due to the contacts made with individuals in patch 2 by 20% of those in patch 1, since patch 2 has higher transmission and shedding rates. These contacts would not have occurred if there was no movement. Similarly, the epidemic in patch 2 is noticed to decrease for non-symmetric mixing compared to when there is no movement, as a result of 30% of individuals in patch 2 mixing with only those in patch 1. Overall, our results show that movement between human populations may lead to an increase or decrease in the epidemic size depending on the outbreak situation and infection rate in the home or destination patch. In addition, they show that movement may be allowed from the population with high transmission and shedding rate to the other populations with lower rates, but not vice versa.

To study the impact of the diffusing pathogens on the epidemic, we consider two scenarios: when an outbreak starts with infected individuals only and when the outbreak starts with only diffusing pathogens. The latter scenario can be seen as a scenario where pathogens diffuse to completely susceptible populations from infectious populations. Our numerical simulations show that the epidemic takes off faster when the outbreak begins with diffusing pathogens compared to when it begins with infected individuals. Epidemic take-off seems delayed when the outbreak begins with infected individuals because of the time needed to shed enough pathogens that will further infect others. We also explore how the diffusion rate of pathogens impacts the disease dynamics. We solved the reduced system of ODEs and the coupled PDE–ODE model with small, moderate, and high diffusion rates of pathogens. Our results show that an increase in the diffusion rate of pathogens leads to a lower epidemic peak, and an increase in the epidemic time. Having a high diffusion rate, which may be interpreted as having a windy situation that blows the pathogens around in the air randomly, may lead to the pathogens being blown away from the settlements by the wind, there by leading to a decrease in cases. Although the analysis used to obtain our reduced system of ODEs is only valid in the limit of fast diffusing pathogens, we observe from our numerical simulations that the results obtained using the ODE model agree well with those of the full coupled PDE–ODE model, even for small diffusion rates of pathogens. This suggests that the reduced ODE system can be used to study the disease dynamics instead of the more complicated PDE–ODE model. An important and unique feature of the ODE system is that it includes a diffusion parameter, which can be used to study the effect of diffusion of pathogen on the disease dynamics.

Summarily, the numerical simulations for both the coupled model and the reduced system of ODEs predict a change in the epidemic peak size and time with human mobility and diffusion. Our results show a decrease in the epidemic peak and an increase in the epidemic time as the diffusion rate of pathogens increases in both the coupled PDE–ODE model and the reduced system of ODEs. In the absence of movement, the epidemic is higher in the patch with higher transmission and shedding rates. In addition, when infections start with no pathogens in the air, the model predicts a delay in the epidemic take-off time relative to when infections start with pathogens, and this delay increases as the diffusion rate of pathogens increases. Our primary results show that accounting for human mobility in a heterogeneous environment is a great way to account for diseases spread in this heterogeneous world, and it is indeed worth considering. We have a similar result with David et al. (2020) when there is no human mobility between patches, and Bichara et al. (2015), where they considered only direct transmission pathway.

There are several future directions to the modeling framework presented in this paper. In this study, we assume that infections can only take place when individuals are in the patches and not outside the population patches (in the bulk region). An interesting extension of this work is to allow for infections to take place in the bulk region, which will change the model to a reaction-diffusion system. It would also be worthwhile to extend the model to include direct transmission of infections from host–host. We have considered only the leading-order terms for the reduced ODE system in this study. It would be interesting to include the \({\mathcal {O}}(\nu )\) terms that incorporate the effect of the location of patches into the ODE systems. In this instance, the basic reproduction number and final size relation derived from the ODE system will depend on the location of the patches. It is straightforward to extend our current model to an n-patch model to study the impact of human mobility on disease prevalence among multiple regions. Another interesting future direction is to include interventions such as vaccination and treatment, especially in an extreme mobility scenario. Furthermore, considering the unpredictable and stochastic effect from the wind over time, a limitation of our study includes the inability to explicitly model the diffusion of pathogens in the population patches. We know that in real-world scenarios, pathogens would need to diffuse into the population before infections can take place and this diffusion process may be non-constant. We accounted for this limitation by giving scenarios of changes in the diffusion rate of pathogens and measure their effect on the disease dynamics. A possible future work in this direction will be to adapt a time-dependent diffusion term using some stochastic processes. Despite these limitations, our modeling framework provides insights into the potential impact of human mobility on the dynamics of airborne diseases. Our results give insights on boarder control (closing and opening of boarders) between two regions during a pandemic, such as COVID-19 (Yuan et al. 2022). Our study serves as a foundation for other studies and its framework can be applied to modeling other infectious diseases.