Predicting the burden of co-infections in seasonally driven dynamics of pediatric rotavirus and norovirus

Abstract

Rotavirus and norovirus are principal viral agents of acute gastroenteritis, primarily transmitted through close contact. Although each virus has its own capability to spread the epidemic, rotavirus and norovirus infection simultaneously is known to have more serious repercussions for children. To examine this coalition, we construct a robust co-infection mathematical model to analyze the prevalence of these diseases and the factors influenced by varying seasonal transmission rates. We conduct numerical simulations and equilibria analyses to capture the dynamics of the individual diseases and their co-infections. We investigate basic and seasonal reproductive numbers, perform sensitivity analysis on key parameters, and numerically explore shifts in the timing and strength of seasonal transmission rates. The research demonstrates how seasonal dynamics significantly impact reproductive numbers, as well as drive the potential burden of co-infection.

1. Introduction

There are several pediatric diseases, such as influenza, diarrhea, pneumonia, Hepatitis A, which represent serious burdens on public health globally, especially among children under the age of five, who are more at risk of a number of contagious and non-contagious illnesses (O'Ryan & Matson, 1990; Shapiro & Hadler, 1990; Doyle, 1976; Strangert, 1976; Bartlett et al., 1985). These diseases contribute significantly to high mortality and morbidity cases, particularly in areas with limited access to healthcare facilities. Many of these infections follow seasonal patterns, influenced by climatic conditions and social factors (Pieter and Heikkinen, 2011; Suryadevara & Domachowske, 2021). The impact of seasonality on childhood diseases is an important aspect to consider in order to gain a better understanding of effective prevention and intervention strategies (Brewster & Greenwood, 1993; Shah et al., 2006).

Each day, millions of children encounter the critical challenges of co-infections of different diseases. The intensity of illness resulting from the simultaneous occurrence of two or more diseases is especially severe among children with immature immune systems (Thorburn et al., 2006). Co-infections have the capacity to aggravate the magnitude of disease and further worsen treatment outcomes. For instance, co-infections arising from viruses and bacterial pathogens often yields acute clinical cases in comparison to single infectious agents (Bakaletz, 2017; Mathew et al., 2019). Similarly, co-infections from gastrointestinal diseases, in particular those involving rotavirus and norovirus, can result in severe diarrhea and higher rates of hospitalization (De Grazia et al., 2020).

The interrelation of dual infections and disease seasonality can add complexities to disease transmission dynamics (Tanner et al., 2012; Fillatre et al., 2018). Due to the high prevalence of wide-ranging pathogens during peak seasons, there are broad chances of multiple co-infections, amplifying risks for young children. This highlights the necessity for a comprehensive study of the processes involved in mixed infections and their related risk determinants. Here, we design and assess a model of two contagious diseases with the possibility of co-infection and investigate any effects derived from seasonality.

Amidst the numerous childhood diseases, rotavirus and norovirus are regarded as major contributors of gastrointestinal infections (GI), specifically in infants and young children (Manatsathit et al., 2002; Riera-Montes et al., 2018). Rotavirus is considered as an ubiquitous pathogen in the world of contagious GI (Margaret et al., 2009). It is renowned for its exceptional resilience and capability for swift transmission (Parashar et al., 2003). Almost every child gets infected with this virus at least one time before the age of five and the main transmission pathway is through faecal-oral transmission (White et al., 2008; Malek et al., 2006; Hochwald & Kivela, 1999). Before the development of the rotavirus vaccine in the 2000s, it caused nearly 3 million cases of acute gastroenteritis in children, almost 60,000 hospitalizations and around 37 deaths each year in the United States of America (Fischer et al., 2007; Malek et al., 2006).

Maternal antibodies developed during gestation act as a defense mechanism against rotavirus among newborns (Raúl Velázquez et al., 2000). This is how they are temporarily protected during the early months of life. However, soon these maternal antibodies becomes weaker, making infants confront the virus with their growing immune systems. They may experience acute manifestations of the infection due to their developing immune systems. Once infected and recovered, patients gain temporary immunity, however there is still a chance of being reinfected at any age. As children move into the toddler years, immunity becomes stronger, which contributes to less severity in the subsequent reinfections (Ward & Bernstein, 1994). On the contrary, adults generally have a higher level of already existed immunity by experiencing rotavirus during childhood (Kapikian et al., 1983).

Just like rotavirus, norovirus is a highly contagious disease related to gastroenteritis (Ahmed et al., 2014). It causes symptoms such as diarrhea, vomiting, and nausea. It has been identified as the “winter vomiting disease” due to it seasonality (Rohayem, 2009). Anyone can get infected with norovirus. It is a common illness which usually spreads through contact or contaminated food or surfaces and the outbreaks occur mostly in colder months (Ahmed et al., 2013). Symptoms among children and adults are usually the same. However, adults may experience more diarrhea than children and children may vomit more than adults.

The prevalence of rotavirus and norovirus can be seen in cooler months, and reduces as temperatures rise. They both occur year-round but exhibit notable seasonality during the winter season (Levy et al., 2009; Mans, 2019). Rotavirus and norovirus are both leading causes of acute gastroenteritis, which remains a major health concern (Manatsathit et al., 2002; Riera-Montes et al., 2018). Each virus alone has the potential to cause acute intestinal inflammation, leading to severe dehydration and hospitalization cases (World Health Organization, 2013; Calderwood et al., 2022). However, co-infection worsens these perils in a noticeable manner. Studies have shown that the involvement of both viruses simultaneously may result in severe diarrhea and vomiting among children (Yang et al., 2022; Mikounou Louya et al., 2019).

Extreme weather changes, such as cold winters, excessive heat and temperature variations, are influential factors for the prevalence of infectious agents related diseases (Sumi et al., 2013; Kraay et al., 2018; Wang et al., 2018). Both rotavirus and norovirus can be attributed to a variety of environmental drivers, including temperature and humidity. Such climate changes are predicted to directly influence the outbreaks of both viruses. Winter temperatures tend to increase their transmission, while summer temperatures either reduce transmission or have no significant effect. It is important to note that norovirus outbreaks exhibit clear seasonality, with a single annual peak during winter and early spring (around December to February) in most temperate regions (Ahmed et al., 2013). Studies also highlight a similar pattern for rotavirus (Patel et al., 2013), indicating a regular, once-per-year peak in incidence during the winter season. These uniform and steady trends in seasonality, manifesting symmetric annual oscillations in transmission, motivate the use of a simple sinusoidal function to represent seasonal forcing in disease propagation models. Furthermore, the regular patterns of these epidemic curves support the use of cosine forcing to explore disease dynamics on a broader scale. While the observed periodic trends motivates the simplified cosine approach we use here, more complex or asymmetric functions could be used to capture additional seasonality. Mathematical modeling can aid in predicting disease occurrence and assist public health organizations in developing effective strategies. They also contribute in preparing evidence-based protocols through the collaboration of epidemiologists, mathematicians, virologists, and public health authorities.

Existing mathematical models of rotavirus and norovirus distribute the population into compartments according to their level of infection intensity considering Susceptible-Infected-Recovered (SIR) (Chandran et al., 2006; Freiesleben de Blasio et al., 2010; Parashar et al., 2006) and Susceptible-Exposed-Infected-Recovered (SEIR) (Asare et al., 2020; Gaythorpe et al., 2018; Lopman et al., 2014; Shuaib and Riyapan, 2020; Van Effelterre et al., 2010; Vanderpas et al., 2009) frameworks. However these model focus on rotavirus or norovirus individually and do not consider possible scenarios of co-infection. On the other hand, co-infection dynamics have been incorporated into mathematical theory of various diseases, such as, listeriosis and anthrax diseases (Osman & Daniel Makinde, 2018), HIV and malaria (Mukandavire et al., 2009), COVID-19 and tuberculosis (Getnet Mekonen & Obsu, 2022). However, models of co-infection of rotavirus and norovrus, as well as research on the implications of their co-infection are still in its preliminary phase.

In this study, we develop a novel and complex mathematical model for rotavirus and norovirus co-infection in a day-care setting. We investigate these two pathogens by capturing the fundamental components of their transmission, including their seasonal patterns and the risks of co-infections. By parameterizing our model for rotavirus and norovirus, we explore the influence of varying transmission rates on basic and seasonal reproductive numbers. In seasonal environments, our model tracks the number of co-infected cases, and highlights the potential difference between the average and maximum number of cases during seasonal peaks. In particular, low baseline transmission rates with high seasonal variations can lead to larger than expected peaks in the number of infected cases for both rotavirus and norovirus as well as for co-infections.

2. Base model development

Rotavirus and norovirus are principal viral agents of acute gastroenteritis mainly transmitted by close contact. Both of these infectious diseases have their own capabilities for spreading. However, what further amplifies the severity is the potential for co-infection by both rotavirus and norovirus. Cases of co-infection are known to have more serious repercussions in children. To examine the coalition of the two viruses simultaneously, we developed an epidemiological model tracking the spread of rotavirus, norovirus, and co-infection of both diseases. Below, we describe the biological assumptions and details of the model development. A flow diagram overview of the model is shown in Fig. 1 and parameters are described in Table 1.

Fig. 1.

Flow diagram of the Co-infected model.

Table 1.

Parameter Table. Values listed were used throughout numerical simulations and their listed range was used as bounds in the parameter sensitivity analysis. Rota stands for rotavirus, Noro stands for norovirus, and co-inf stands for co-infection of rotavirus and norovirus.

Parameter	Symbol	Value	Range	Source
Natural birth rate	λ	0.0292	[0.0029, 0.29]	assumed
Rota transmission rate	βR	0.7–2	[0.7, 5]	Lawrence Omondi et al. (2015)
Noro transmission rate	βN	1.5–3.47	[1.5, 5]	Steele et al. (2020)
Natural death rate	ν	1/(75 × 365)	[0.18, 0.54] × 10−4	assumed
Rota immunity loss rate	δR	0.01	[0.005, 0.015]	assumed
Noro immunity loss rate	δN	0.01	[0.005, 0.015]	assumed
Co-inf immunity loss rate	δRN	0.01	[0.005, 0.015]	assumed
Noro symptom development rate	θN	0.76	[0.5, 1]	Vanderpas et al. (2009)
Rota induced death rate	τR	0.01	[0.005, 0.15]	assumed
Noro induced death rate	τN	0.01	[0.005, 0.15]	assumed
Rota recovery rate	γR	1/5	[0.1, 0.5]	Lappe et al. (2023)
Noro recovery rate	γN	0.64561	[0.1, 1]	Vanderpas et al. (2009)
Recovery rate of co-inf	γRN	1/5	[0.1, 0.5]	assumed
Rota seasonal amplitude	ϵ1	0–1	[0.01, 1]	assumed
Noro seasonal amplitude	ϵ2	0–1	[0.01, 1]	assumed
Rota seasonal shift	p	0–365/2	[0.01, 365/2]	assumed
Noro seasonal shift	q	0–365/2	[0.01, 365/2]	assumed

The model divides the total population into eleven compartments with individuals that are susceptible (S), exposed to norovirus and are infected but not yet infectious (E_N), infectious with norovirus (I_N), recovered from norovirus (R_N), infected and infectious with rotavirus (I_R), infected with rotavirus and exposed to norovirus (E_RN), co-infected (I_RN), recovered from rotavirus (R_R), exposed to norovirus having rotavirus immunity ($E_N^R$), infected with norovirus having rotavirus immunity ($I_N^R$), infected with rotavirus having norovirus immunity ($I_R^N$) and recovered from co-infection (R_RN). Taking this into consideration, the total population denoted by N is given by

$$N=S+E_N+I_N+R_N+I_R+E_{RN}+I_{RN}+R_R+E_N^R+I_N^R+I_R^N+R_{RN}$$ (1)

We assume the susceptible population is recruited at the constant natural birth rate, denoted by λ (black arrow, Fig. 1) and all individuals undergo natural mortality rate ν (blue arrows, Fig. 1). The infection process between susceptible and the infectious individuals is influenced by density-dependent transmission. Susceptible individuals are infected by norovirus through contact with infectious individuals at transmission rate β_N. Norovius has a latent stage when infected individuals are initially asymptomatic and not yet infectious (Gaythorpe et al., 2018), which we track in the exposed compartments. The transmission of norovirus to susceptible individuals moves them to a norovirus-infected latent stage, where the individual is infected but not yet infectious, takes for following form:

$$\frac{{\beta }_{N}S(I_N+I_{RN}+I_N^R)}{N}.$$

Here we assume norovirus transmission rates are similar for those who are infectious with norovirus, regardless if they are currently co-infected with rotavirus (I_RN) or recovered from rotavirus ($I_N^R$). The transmission of rotavirus to a susceptible individual moves them to a rotavirus infected stage (without going through a latent stage) taking the following form:

$$\frac{{\beta }_{R}S(I_R+I_{RN}+I_R^N)}{N}.$$

Here, we take similar assumptions as above and assume all individuals infected with rotavirus have the same transmission rates regardless of their status with respect to norovirus. Individuals exposed and infected with norovirus can get infected with rotavirus with the following transmission rate:

$$\frac{{\beta }_R{E}_N(I_R+I_{RN}+I_R^N)}{N}+\frac{{\beta }_R{I}_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}.$$

Individuals who have recovered from norovirus and gained temporary immunity can get infected with rotavirus at transmission rate:

$$\frac{{\beta }_R{R}_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}.$$

Individuals infected with rotavirus can enter the norovirus exposed class at transmission rate:

$$\frac{{\beta }_N{I}_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}.$$

Finally, those individuals who have recovered from rotavirus and gained temporary immunity can also get infected with norovirus at transmission rate:

$$\frac{{\beta }_N{R}_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}.$$

These transmission terms are shown as red arrows in Fig. 1.

Individuals infected with norovirus in the exposed class will develop clinical symptoms at rate θ_N, and become infectious with norovirus (orange arrow, Fig. 1). These infected individuals may die at disease induced mortality rate of τ_N (blue arrow, Fig. 1) or recover at rate γ_N (brown arrow, Fig. 1). These recovered individuals lose their temporary immunity at rate δ_N (green arrow, Fig. 1) making them susceptible again. Similarly, individuals infected with rotavirus may die at disease-induced mortality rate τ_R (blue arrow, Fig. 1) or recover at rate γ_R (brown arrow, Fig. 1). These recovered individuals lose their temporary immunity at rate δ_R (green arrow, Fig. 1), and hence return to the susceptible class.

Since our model considers the possibility of experiencing co-infection, the individuals from the I_R class upon being exposed with norovirus infection can go to the E_RN class from where they further progress to the I_RN i.e. co-infected class. In the same way, individuals from E_N and I_N compartments can move to I_RN class when infected with rotavirus. Moreover, individuals recovered from norovirus who have acquired temporary immunity, may encounter an infected rotavirus person. Such individual will move to the $I_R^N$ class. Likewise, individuals recovered from rotavirus who have acquired temporary immunity may encounter an infectious norovirus person. Such individual may move to the $E_N^R$ class and subsequently progressing to the $I_N^R$ class after showing clinical symptoms. The infected individuals with co-infection recover at the rate of γ_RN. Furthermore, we assume that recovered individuals move to the susceptible compartment after getting their temporary immunity wanes at rate δ_RN. Putting this together, we have the system of equations as follows:

$$\begin{array}{l}\frac{dS}{dt}=\lambda -\frac{{\beta }_{N}S(I_N+I_{RN}+I_N^R)}{N}-\frac{{\beta }_{R}S(I_R+I_{RN}+I_R^N)}{N}-\nu S+{\delta }_N{R}_N+{\delta }_R{R}_R+{\delta }_{RN}{R}_{RN}\end{array}$$ (2a)

$$\begin{array}{l}\frac{d{E}_N}{dt}=\frac{{\beta }_{N}S(I_N+I_{RN}+I_N^R)}{N}-\frac{{\beta }_R{E}_N(I_R+I_{RN}+I_R^N)}{N}-\nu E_N-{\theta }_N{E}_N\end{array}$$ (2b)

$$\frac{d{I}_N}{dt}={\theta }_N{E}_N-\nu I_N-{\tau }_N{I}_N-{\gamma }_N{I}_N-\frac{{\beta }_R{I}_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}$$ (2c)

$$\frac{d{R}_N}{dt}={\gamma }_N{I}_N-\nu R_N-{\delta }_N{R}_N-\frac{{\beta }_R{R}_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}$$ (2d)

$$\frac{d{I}_R}{dt}=\frac{{\beta }_{R}S(I_R+I_{RN}+I_R^N)}{N}-\frac{{\beta }_N{I}_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}-\nu I_R-{\tau }_R{I}_R-{\gamma }_R{I}_R$$ (2e)

$$\frac{d{E}_{RN}}{dt}=\frac{{\beta }_N{I}_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}-{\theta }_N{E}_{RN}-\nu E_{RN}$$ (2f)

$$\frac{d{I}_{RN}}{dt}=\frac{{\beta }_R{E}_N(I_R+I_{RN}+I_R^N)}{N}+\frac{{\beta }_R{I}_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}+{\theta }_N{E}_{RN}-\nu I_{RN}-{\tau }_R{I}_{RN}-{\tau }_N{I}_{RN}-{\gamma }_{RN}{I}_{RN}$$ (2g)

$$\frac{d{R}_R}{dt}={\gamma }_R{I}_R-\nu R_R-{\delta }_R{R}_R-\frac{{\beta }_N{R}_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}$$ (2h)

$$\frac{d{E}_N^R}{dt}=\frac{{\beta }_N{R}_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}-{\theta }_N{E}_N^R-\nu E_N^R$$ (2i)

$$\frac{d{I}_N^R}{dt}={\theta }_N{E}_N^R-\nu I_N^R-{\tau }_N{I}_N^R-{\gamma }_N{I}_N^R$$ (2j)

$$\frac{d{I}_R^N}{dt}=\frac{{\beta }_R{R}_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}-\nu I_R^N-{\tau }_R{I}_R^N-{\gamma }_R{I}_R^N$$ (2k)

$$\frac{d{R}_{RN}}{dt}={\gamma }_N{I}_N^R+{\gamma }_R{I}_R^N-{\delta }_{RN}{R}_{RN}+{\gamma }_{RN}{I}_{RN}$$ (2l)

Parameter values are summarized in Table 1. Natural birth and death rates, λ and ν are parameterized for a disease-free population size of 800 people. Baseline transmission rates for rotavirus, β_R, and norovirus, β_N were parameterized so the respective basic reproductive numbers are in the same order of magnitude as previous modeling efforts (Lawrence Omondi et al., 2015; Steele et al., 2020). We further explore a range around these baseline values. The duration of rotavirus infection has been estimated to be about 5 days (Lappe et al., 2023), so we parameterize the recovery rate γ_R = 1/5. We assumed the recovery rate of co-infected cases γ_RN is similar. Previous models of norovirus were used to estimate the rate of developing symptoms θ_N and recovery rate γ_N (Vanderpas et al., 2009). The remaining parameter values listed in Table 1 were assumed due to the lack of data. Hence, we conduct a sensitivity analysis to identify the influential ones in Sections 3.1.1, 3.2.1.

3. Base model analysis

For a preliminary analysis of the co-infection model, we first study the characteristics of the sub-models for rotavirus and norovirus. We examine disease-free equilibria and calculate basic reproductive numbers.

3.1. Rotavirus only sub-model

Considering only the rotavirus infections from the co-infection model, the sub-model of rotavirus is:

$$\frac{dS}{dt}=\lambda -\frac{{\beta }_{R}S{I}_R}{N}-\nu S+{\delta }_R{R}_R$$ (3a)

$$\frac{d{I}_R}{dt}=\frac{{\beta }_{R}S{I}_R}{N}-\nu I_R-{\tau }_R{I}_R-{\gamma }_R{I}_R$$ (3b)

$$\frac{d{R}_R}{dt}={\gamma }_R{I}_R-\nu R_R-{\delta }_R{R}_R$$ (3c)

Theorem 3.1

The disease-free equilibrium point, $X_0^R=\left(\frac{\lambda }{\nu },0,0\right)$ is unstable if ${\mathcal{R}}_0^R=\frac{{\beta }_R}{\nu +{\tau }_R+{\gamma }_R}>1$.

The proof is in Appendix A.1.

Theorem 3.2

The endemic equilibrium point, $X_E^R=(S_E^R,I_E^R,R_E^R)$ is positive if ${\mathcal{R}}_0^R=\frac{{\beta }_R}{\nu +{\tau }_R+{\gamma }_R}>1$.

The proof is in Appendix A.1 and stability is shown numerically.

3.1.1. Sensitivity analysis of rotavirus only sub-model

We conduct a parameter sensitivity analysis of the rotavirus sub-model and assessing the influence of each parameter on the basic reproductive number:

$${\mathcal{R}}_0^R=\frac{{\beta }_R}{\nu +{\tau }_R+{\gamma }_R}$$

The sensitivity for a given parameter, say p, is defined as how the model behaves to a slight change in the parameter value (Martcheva, 2015), and is given by:

$$S_p=\frac{\partial {\mathcal{R}}_0^R}{\partial p}\frac{p}{{\mathcal{R}}_0^R}.$$

For the parameters of rotavirus only sub model, we have

$$\begin{array}{l}{S}_{\alpha }=\frac{\partial {\mathcal{R}}_0^R}{\partial \lambda }\frac{\lambda }{{\mathcal{R}}_0^R}=0\\ S_{{\beta }_R}=\frac{\partial {\mathcal{R}}_0^R}{\partial {\beta }_R}\frac{{\beta }_R}{{\mathcal{R}}_0^R}=1\\ S_{\nu }=\frac{\partial {\mathcal{R}}_0^R}{\partial \nu }\frac{\nu }{{\mathcal{R}}_0^R}=\frac{-\nu }{(\nu +{\tau }_R+{\gamma }_R)}\\ S_{{\delta }_R}=\frac{\partial {\mathcal{R}}_0^R}{\partial {\delta }_R}\frac{{\delta }_R}{{\mathcal{R}}_0^R}=0\\ S_{{\tau }_R}=\frac{\partial {\mathcal{R}}_0^R}{\partial {\tau }_R}\frac{{\tau }_R}{{\mathcal{R}}_0^R}=\frac{-{\tau }_R}{(\nu +{\tau }_R+{\gamma }_R)}\\ S_{{\gamma }_R}=\frac{\partial {\mathcal{R}}_0^R}{\partial {\gamma }_R}\frac{{\gamma }_R}{{\mathcal{R}}_0^R}=\frac{-{\gamma }_R}{(\nu +{\tau }_R+{\gamma }_R)}\end{array}$$

The sensitivity analysis of the rotavirus sub-model (Table 2) displays that β_R has the highest in magnitude sensitivity index and therefore is the most influential parameter. Furthermore positive changes in β_R affects the spread of the virus positively. On the other hand, positive changes in less influential parameters γ_R, ν, and τ_R negatively influence disease spread.

Table 2.

Sensitivity indices of ${\mathcal{R}}_0^R$ evaluated using the parameter values in Table 1.

Parameter	Symbol	Sensitivity indices
Transmission rate of rotavirus	βR	1
Natural death rate	ν	−0.000174
Disease induced mortality rate	τR	−0.0476107
Recovery rate in adults	γR	−0.952

3.2. Norovirus only sub-model

If we consider only the presence of norovirus disease, the system is reduced to the following set of equations:

$$\frac{dS}{dt}=\lambda -\frac{{\beta }_{N}S{I}_N}{N}-\nu S+{\delta }_N{R}_N$$ (4a)

$$\frac{d{E}_N}{dt}=\frac{{\beta }_{N}S{I}_N}{N}-\nu E_N-{\theta }_N{E}_N$$ (4b)

$$\frac{d{I}_N}{dt}={\theta }_N{E}_N-\nu I_N-{\tau }_N{I}_N-{\gamma }_N{I}_N$$ (4c)

$$\frac{d{R}_N}{dt}={\gamma }_N{I}_N-\nu R_N-{\delta }_N{R}_N$$ (4d)

Theorem 3.3

The disease-free equilibrium point, $X_0^N=\left(\frac{\lambda }{\nu },0,0,0\right)$ is unstable if ${\mathcal{R}}_0^N=\frac{{\beta }_N{\theta }_N}{(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)}>1$.

The proof is in appendix Appendix A.2.

Theorem 3.4

The endemic equilibrium point, $X_E^N=(S_E^N,I_E^N,R_E^N)$ is positive if ${\mathcal{R}}_0^N=\frac{{\beta }_N{\theta }_N}{(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)}>1$.

The proof is in Appendix A.2 and stability is shown numerically.

3.2.1. Sensitivity analysis of norovirus only sub-model

Like the parameter sensitivity study for the rotavirus sub-model, we compute the sensitivity analysis on the Norovirus basic reproductive number,

$${\mathcal{R}}_0^N=\frac{{\beta }_N{\theta }_N}{(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)}$$

for parameters of the norovirus sub-model:

$$\begin{array}{l}{S}_{\alpha }=\frac{\partial {\mathcal{R}}_0^N}{\partial \lambda }\frac{\lambda }{{\mathcal{R}}_0^N}=0\\ S_{{\beta }_N}=\frac{\partial {\mathcal{R}}_0^N}{\partial {\beta }_N}\frac{{\beta }_N}{{\mathcal{R}}_0^N}=1\\ S_{\nu }=\frac{\partial {\mathcal{R}}_0^N}{\partial \nu }\frac{\nu }{{\mathcal{R}}_0^N}=\frac{-\nu (2\nu +{\tau }_N+{\gamma }_N+{\theta }_N)}{(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)}\\ S_{{\delta }_N}=\frac{\partial {\mathcal{R}}_0^N}{\partial {\delta }_N}\frac{{\delta }_N}{{\mathcal{R}}_0^N}=0\\ S_{{\theta }_N}=\frac{\partial {\mathcal{R}}_0^N}{\partial {\theta }_N}\frac{{\theta }_N}{{\mathcal{R}}_0^N}=\frac{-\nu }{\nu +{\theta }_N}\\ S_{{\tau }_N}=\frac{\partial {\mathcal{R}}_0^N}{\partial {\tau }_N}\frac{{\tau }_N}{{\mathcal{R}}_0^N}=\frac{-{\tau }_N}{(\nu +{\tau }_N+{\gamma }_N)}\\ S_{{\gamma }_N}=\frac{\partial {\mathcal{R}}_0^N}{\partial {\gamma }_N}\frac{{\gamma }_N}{{\mathcal{R}}_0^N}=\frac{-{\gamma }_N}{(\nu +{\tau }_N+{\gamma }_N)}\end{array}$$

The sensitivity analysis of the norovirus sub-model shows that increases in parameters β_N influence the spread of the virus in a positive way whereas, increases in the remaining parameters, such as ν, θ_N,τ_N and γ_N have a negative affect on disease spread. For the magnitudes of the impact of the parameters, refer to Table 3. While the parameters τ_R, τ_N, γ_R, γ_N, and θ_N are uncertain and presumed, our sensitivity analysis shows that they are relatively small, making them less impactful. therefore, we focus on varing the transmission rates β_R and β_N in our numerical analyses below.

Table 3.

Sensitivity indices of ${\mathcal{R}}_0^N$ evaluated using the parameter values in Table 1.

Parameter	Symbol	Sensitivity indices
Transmission rate of norovirus	βN	1
Natural death rate	ν	−0.000104
Rate of developing symptoms	θN	−0.000048
Disease induced mortality rate	τN	−0.01525
Recovery rate in adults	γN	−0.9846

3.3. Co-infection model

Moving to the full model, we calculated the disease-free equilibrium for the co-infection which is given by

$$X_0^C=\left(\frac{\lambda }{\nu },0,0,0,0,0,0,0,0,0,0,0\right)$$

We evaluate the basic reproductive number $({\mathcal{R}}_0^C)$ for the full model using the Next Generation Matrix method (Van den Driessche & Watmough, 2002). The sub-model involving only the infectious compartments are as follows:

$$\begin{array}{l}\frac{d{E}_N}{dt}=\frac{{\beta }_{N}S(I_N+I_{RN}+I_N^R)}{N}-\frac{{\beta }_R{E}_N(I_R+I_{RN}+I_R^N)}{N}-\nu E_N-{\theta }_N{E}_N\end{array}$$ (5a)

$$\frac{d{I}_N}{dt}=-\frac{{\beta }_R{I}_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}+{\theta }_N{E}_N-\nu I_N-{\tau }_N{I}_N-{\gamma }_N{I}_N$$ (5b)

$$\begin{array}{l}\frac{d{I}_R}{dt}=\frac{{\beta }_{R}S(I_R+I_{RN}+I_R^N)}{N}-\frac{{\beta }_N{I}_R\left(I_N+I_{RN}+I_N^R\right)}{N}-\nu I_R-{\tau }_R{I}_R-{\gamma }_R{I}_R\end{array}$$ (5c)

$$\frac{d{E}_{RN}}{dt}=\frac{{\beta }_N{I}_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}-{\theta }_N{E}_{RN}-\nu E_{RN}$$ (5d)

$$\begin{array}{l}\frac{d{I}_{RN}}{dt}=\frac{{\beta }_R{E}_N(I_R+I_{RN}+I_R^N)}{N}+\frac{{\beta }_R{I}_N\left(I_R+I_{RN}+I_R^N\right)}{N}+{\theta }_N{E}_{RN}-\nu I_{RN}-{\tau }_R{I}_{RN}-{\tau }_N{I}_{RN}-{\gamma }_{RN}{I}_{RN}\end{array}$$ (5e)

$$\frac{d{E}_N^R}{dt}=\frac{{\beta }_N{R}_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}-{\theta }_N{E}_N^R-\nu E_N^R-{\tau }_N{E}_N^R$$ (5f)

$$\frac{d{I}_N^R}{dt}={\theta }_N{E}_N^R-\nu I_N^R-{\tau }_N{I}_N^R-{\gamma }_N{I}_N^R$$ (5g)

$$\frac{d{I}_R^N}{dt}=\frac{{\beta }_R{R}_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}-\nu I_R^N-{\tau }_R{I}_R^N-{\gamma }_R{I}_R^N$$ (5h)

In vector form, this is written as:

$$\frac{d\vec{x}}{dt}=\mathbb{F}(\vec{x})-\mathbb{V}(\vec{x})$$ (6)

where $\vec{x}$ is the vector of all infectious compartments [$E_N,I_N,I_R,E_{RN},I_{RN},E_N^R,I_N^R,I_R^N$]. Here, $\mathbb{F}(\vec{x})$ symbolizes the transmission terms of the sub-model and $\mathbb{V}(\vec{x})$ contains all the transition terms of the sub-model. We divide the above sub model into two parts: $\mathbb{F}$ which contains the transmission terms of the infection and $\mathbb{V}$ which contains the transition terms of the model. Let $\mathcal{F}(\vec{x})=\frac{\partial \mathbb{F}(\vec{x})}{\partial \vec{x}}$ and $\mathcal{V}(\vec{x})=\frac{\partial \mathbb{V}(\vec{x})}{\partial \vec{x}}$ i.e. the partial derivatives of $\mathbb{F}$ and $\mathbb{V}$. The Next Generation Matrix is given by ${\mathcal{F}\mathcal{V}}^{-1}$ where ${\mathcal{V}}^{-1}$ is the inverse of matrix $\mathcal{V}$. Finally, the basic reproductive number given by the spectral radius ${\mathcal{R}}_0^C=\rho ({\mathcal{F}\mathcal{V}}^{-1})$.

The co-infection model has two reproductive numbers:

$${\mathcal{R}}_0^R=\frac{{\beta }_R}{\nu +{\tau }_R+{\gamma }_R}$$ (7)

and

$${\mathcal{R}}_0^N=\frac{{\beta }_N{\theta }_N}{(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)}$$ (8)

which are same as the basic reproductive numbers from the rotavirus only and norovirus only models respectively (Appendix A.1) and (A.5)). The basic reproductive number for the full co-infection model is then,

$${\mathcal{R}}_0^C=max\{{\mathcal{R}}_0^R,{\mathcal{R}}_0^N\}$$ (9)

4. Seasonal model extension

Seasonal shifts have been recognized as some of the primary drivers for swift transmission and severe course of infections (Olinky et al., 2008), (Tanaka & Aihara, 2013), (Fine & Clarkson, 1982). Many diseases go through seasonal variations in frequency, severity, and mortality (Fisman, 2007). Seasonal variations might be caused as by climate, temperature and humidity, as well as social behaviors (contact patterns) influenced by seasonal cycles or holidays. Mathematical modeling is a standard and efficient strategy to study the influence of seasonality on epidemic spreads (Roy, 1991), (Aron & Schwartz, 1984), (Bakary et al., 2018), (Dyrdak et al., 2020). Here, we extend our co-infection epidemic model with seasonal forcing functions fueling transmission dynamics. We consider direct contacts rate of rotavirus and norovirus to be seasonal and introduce sinusoidal forcing in the transmission rates.

4.1. Model development

Although there are several types of seasonal forcing (Altizer et al., 2006; Tanaka & Aihara, 2013), we have considered sinusoidal forcing since it is often used in representing seasonal variations in epidemic models (Duarte et al., 2019; Fatema Nipa et al., 2021). We replace the constant transmission rates β_R and β_N with periodic transmission rates of the form:

$${\beta }_R(t)={\beta }_{R0}\left(1+{ϵ}_{1}cos\left(\frac{2\pi (t+p)}{365}\right)\right)$$ (10)

$${\beta }_N(t)={\beta }_{N0}\left(1+{ϵ}_{2}cos\left(\frac{2\pi (t+q)}{365}\right)\right)$$ (11)

where β_R0 and β_N0 are the mean transmission rates. Parameters ϵ₁ and ϵ₂ correspond to the amplitude of the seasonality whose value lies between zero and unity. The parameters p and q represent the peak shift in the transmission rates and determine when the peak transmission occurs during the year. Here, β_R0 and β_N0 are positive and, time (t) is expressed in days.

Hence, the seasonally forced co-infection model is expressed as

$$\begin{array}{l}\frac{dS}{dt}=\lambda -\frac{{\beta }_N\left(t\right)S(I_N+I_{RN}+I_N^R)}{N}-\frac{{\beta }_R\left(t\right)S(I_R+I_{RN}+I_R^N)}{N}-\nu S+{\delta }_N{R}_N+{\delta }_R{R}_R+{\delta }_{RN}{R}_{RN}\end{array}$$ (12a)

$$\begin{array}{l}\frac{d{E}_N}{dt}=\frac{{\beta }_N\left(t\right)S(I_N+I_{RN}+I_N^R)}{N}-\frac{{\beta }_R\left(t\right)E_N(I_R+I_{RN}+I_R^N)}{N}-\nu E_N-{\theta }_{N}E\end{array}$$ (12b)

$$\frac{d{I}_N}{dt}={\theta }_N{E}_N-\nu I_N-{\tau }_N{I}_N-{\gamma }_N{I}_N-\frac{{\beta }_R(t)I_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}$$ (12c)

$$\frac{d{R}_N}{dt}={\gamma }_N{I}_N-\nu R_N-{\delta }_N{R}_N-\frac{{\beta }_R(t)R_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}$$ (12d)

$$\frac{d{I}_R}{dt}=\frac{{\beta }_R\left(t\right)S(I_R+I_{RN}+I_R^N)}{N}-\frac{{\beta }_N\left(t\right)I_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}-\nu I_R-{\tau }_R{I}_R-{\gamma }_R{I}_R$$ (12e)

$$\begin{array}{l}\frac{d{E}_{RN}}{dt}=\frac{{\beta }_N\left(t\right)I_R\left(I_N+I_{RN}+I_N^R\right)}{N}-{\theta }_N{E}_{RN}-\nu E_{RN}\end{array}$$ (12f)

$$\frac{d{I}_{RN}}{dt}=\frac{{\beta }_R\left(t\right)E_N(I_R+I_{RN}+I_R^N)}{N}+\frac{{\beta }_R\left(t\right)I_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}+{\theta }_N{E}_{RN}-\nu I_{RN}-{\tau }_R{I}_{RN}-{\tau }_N{I}_{RN}-{\gamma }_{RN}{I}_{RN}$$ (12g)

$$\begin{array}{l}\frac{d{R}_R}{dt}={\gamma }_R{I}_R-\nu R_R-{\delta }_R{R}_R-\frac{{\beta }_N\left(t\right)R_R\left(I_N+I_{RN}+I_N^R\right)}{N}\end{array}$$ (12h)

$$\frac{d{E}_N^R}{dt}=\frac{{\beta }_N(t)R_R\left(I_N+I_{\mathit{RN}}+I_N^R\right)}{N}-{\theta }_N{E}_N^R-\nu E_N^R$$ (12i)

$$\frac{d{I}_N^R}{dt}={\theta }_N{E}_N^R-\nu I_N^R-{\tau }_N{I}_N^R-{\gamma }_N{I}_N^R$$ (12j)

$$\frac{d{I}_R^N}{dt}=\frac{{\beta }_R(t)R_N\left(I_R+I_{\mathit{RN}}+I_R^N\right)}{N}-\nu I_R^N-{\tau }_R{I}_R^N-{\gamma }_R{I}_R^N$$ (12k)

$$\frac{d{R}_{RN}}{dt}={\gamma }_N{I}_N^R+{\gamma }_R{I}_R^N-{\delta }_{RN}{R}_{RN}+{\gamma }_{RN}{I}_{RN}$$ (12l)

4.2. Seasonal reproductive numbers

The concept of basic reproductive number has been broadly applied over the years to both autonomous and non-autonomous models of disease transmission (Wesley & Allen, 2009; Bacaër & Guernaoui, 2006; Mitchell & Kribs, 2017). While, the next generation matrix approach was used to calculate ${\mathcal{R}}_0^R$, ${\mathcal{R}}_0^N$, ${\mathcal{R}}_0^C$ in constant environments with constant coefficients, its application to seasonal environments with periodic coefficients needs to be extended. Here, we evaluate two different measures of disease spread in seasonal environments, time-dependent instantaneous reproductive numbers (Bari Hridoy, 2025) and numerically obtained periodic reproductive numbers (Wang & Zhao, 2008).

4.2.1. Instantaneous reproductive numbers

In our co-infection model with seasonal transmission rates of rotavirus and norovirus, the time-dependent seasonal instantaneous reproductive numbers oscillate over time are given by the following equations:

$${\hat{\mathcal{R}}}_0^R(t)=\frac{{\beta }_R(t)}{\nu +{\tau }_R+{\gamma }_R}$$ (13)

and

$${\hat{\mathcal{R}}}_0^N(t)=\frac{{\beta }_N(t){\theta }_N}{(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)}$$ (14)

The co-infected model then has time-dependent seasonal instantaneous reproductive number as:

$${\hat{\mathcal{R}}}_0^C(t)=max\{{\hat{\mathcal{R}}}_0^N(t),{\hat{\mathcal{R}}}_0^R(t)\}$$ (15)

4.2.2. Periodic reproductive numbers

Alongside considering the instantaneous reproductive numbers ${\hat{\mathcal{R}}}_0(t)$, we calculate periodic reproductive numbers ${\tilde{\mathcal{R}}}_0$ using floquet theory following Wang & Zhao, (2008). For the simple SIR model used for rotavirus, the periodic reproductive number ${\tilde{\mathcal{R}}}_0^R$ will correspond with the same value obtained from the case with constant coefficients, ${\mathcal{R}}_0^R$. However, the more complicated SEIR framework for norovirus results in different values for seasonal vs constant environments (Bari Hridoy, 2025; Wang & Zhao, 2008). Here we compute the periodic reproductive number for our norovirus sub-model, where the transmission rate β_N undergoes sinusoidal seasonal forcing throughout the year (Eq. (11)) using methods from Wang & Zhao, (2008) and detailed in Mitchell & Kribs, (2017). We denote the periodic reproductive numbers as:

$${\tilde{\mathcal{R}}}_0^R={\mathcal{R}}_0^R\text{Rotavirus Periodic reproductive number}$$ (16)

$${\tilde{\mathcal{R}}}_0^N\ne {\mathcal{R}}_0^N\text{Norovirus Periodic reproductive number}$$ (17)

Model parameters regulate the reproductive numbers, which are reflected in the epidemiological dynamics of the disease. To explore the role of seasonality in disease transmission, we compare the instantaneous reproductive numbers, the periodic reproductive numbers, and the basic reproductive numbers with constant transmission for both rotavirus and norovirus. The instantaneous reproductive number exhibits oscillations for a short period as a consequence of seasonal factors. The periodic reproductive number gives the average reproductive number as a tool for evaluating epidemic likelihood. Since these viruses tend to soar in colder months, the instantaneous and the periodic reproductive number are useful in capturing seasonal outbreaks accurately. Comparisons between these analytics give us insights into the important roles that seasonality have in driving the dynamics of co-infections in our system.

5. Numerical simulations

In this section, we explore dynamics of our co-infection base model (2) and seasonal model (4.1) with numerical simulations of model solutions and reproductive numbers, alongside heatmaps capturing long term dynamics.

5.1. Base model simulations

The influence of β_R and β_N on the transmission of the individual diseases and as well as when the two viruses persist simultaneously in the base model (2) are shown in Fig. 2. When β_R = 1 and β_N = 1.5, rotavirus infection becomes pervasive (Fig. 2a and 2b). In this case, we also observe short term transient peaks in the proportion of cases that are co-infected (Fig. 2b). Similarly, to observe a case when the norovirus surpasses rotavirus in the spread of the overall disease transmission, we took β_R = 0.7 and β_N = 2.5 shown in Fig. 2c and d. Here, there is a widespread rise in the norovirus as compared to rotavirus. The proportion of co-infected cases also exhibits transient peaks (Fig. 2d). Finally, in Fig. 2e 2f, with β_R = 1.6 and β_N = 3.47, we see a prominence of co-infection cases.

Fig. 2.

Prevalence of rotavirus, norovirus and co-infection (a,c,e) and proportion of infected cases (b,d,f) for three scenarios: 1.) High rotavirus (a,b) with β_R = 1 and β_N = 1.5 where ${\mathcal{R}}_0^R=4.9991$, ${\mathcal{R}}_0^N=0.9682$ and ${\mathcal{R}}_0^C=4.9991$; 2.) High Norovirus (c,d) with β_R = 0.7 and β_N = 2.5 where ${\mathcal{R}}_0^R=3.4994$, ${\mathcal{R}}_0^N=1.6137$ and ${\mathcal{R}}_0^C=3.4994$; and 3.) High co-infection (e,f) with β_R = 1.6 and β_N = 3.47 where ${\mathcal{R}}_0^R=7.9985$, ${\mathcal{R}}_0^N=2.2398$ and ${\mathcal{R}}_0^C=7.9985$.

Variations in the constant transmissions rates of rotavirus β_R and norovirus β_N are shown in Fig. 3. These heatmaps elucidate how the imbalance and variation in these rates influence the basic reproductive numbers (Fig. 3a, 3b and 3c) and the number of individuals infected with rotavirus I_R, norovirus I_N and co-infection I_RN in the long term (Fig. 3d, 3e and 3f). In general, higher values of β_R lead to higher values of ${\mathcal{R}}_0^R$ (yellow regions in Fig. 3a). Likewise, a greater β_N leads to higher values of ${\mathcal{R}}_0^N$ (Fig. 3b). High values of both β_R and β_N correspond with high values of ${\mathcal{R}}_0^C$ (Fig. 3c). The long-term number of rotavirus only infected cases I_R peaks when β_R is high and β_N is low, Fig. 3d. Similarly, we see peaks in the norovirus only cases I_N when β_N is high but β_R is low, Fig. 3e. When both β_R and β_N are increased simultaneously, the number of co-infected cases I_RN dominates, Fig. 3f.

Fig. 3.

Variations in the constant transmission rates, β_R and β_N affect basic reproductive number dynamics for rotavirus (a), norovirus (b), and co-infections (c), and long term dynamics at endemic equilibria for rotavirus (d), norovirus (e), and co-infection cases (f) for base model (2) for 1.5 < t < 3 years. White line on the plots represent where these basic reproductive numbers (7), (8) and (9) equal 1.

5.2. Seasonal model simulations

Variations in the mean of the seasonal transmissions rates of rotavirus β_R0 and norovirus β_N0 are shown in Fig. 4. Here, we explore their influence on the periodic reproductive numbers (Fig. 4a–c), and long-term dynamics of the average infected cases (Fig. 4d–f) and seasonal maximum infected cases (Fig. 4g–i). The periodic reproductive numbers demonstrates the potential of the disease to transmit at any point during the course of the year, governed by seasonal factors. ${\tilde{\mathcal{R}}}_0^R$ is vulnerable to changes in β_R0 because higher value of β_R0 corresponds to higher β_R(t), Fig. 4a. On the other hand, β_N0 drives values of ${\tilde{\mathcal{R}}}_0^N$, Fig. 4b. The cumulative effect of β_R0 and β_N0 on ${\tilde{\mathcal{R}}}_0^C$ is captured in Fig. 4c. The long-term dynamics of the average infected cases (Fig. 4d–f) is similar to those of the non-seasonal base model (Fig. 3d–f). However, the seasonal peaks of long-term dynamics of rotavirus, norovirus and co-infections cases give us new insights (Fig. 4g–i). Here, seasonal peaks of rotavirus only infections are highest for smaller values of β_R0, near the threshold where ${\tilde{\mathcal{R}}}_0^R=1$, white line in Fig. 4g. Similarly, seasonal peaks of norovirus only infections are highest for smaller values of β_N0, near the threshold where ${\tilde{\mathcal{R}}}_0^N=1$, white line in Fig. 4h. Seasonal peaks of co-infection cases occur for large values of β_R0 and β_N0, but we also see a small region with high cases numbers near the thresholds where ${\tilde{\mathcal{R}}}_0^R=1$ and ${\tilde{\mathcal{R}}}_0^N=1$, for moderately low values of β_R0 and β_N0, Fig. 4i. All these heat maps offer clarity into how seasonal variations in transmission rates affect the capability of the disease spread, with ramifications for awareness and control of the virus infection.

Fig. 4.

Variations in the mean of the seasonal transmission rates, β_R0 and β_N0 of equations (10), (11) affect periodic reproductive number dynamics for rotavirus (a), norovirus (b), and co-infections (c); and long term dynamics of average endemic rotavirus (d), norovirus (e), and co-infection cases (f); and the maximum seasonal peak endemic rotavirus (g), norovirus (h), and co-infection cases (i) for the seasonal model (4.1). These long term dynamics were calculated as the respective means and maximums of periodic solutions after transient dynamics for 1.5 < t < 3 years. White line on the plots represent where these periodic reproductive numbers (13), (14) and (15) equal 1.

5.2.1. Strength of seasonality and its effects

Here we explore the strength of seasonality as variations in the amplitude of seasonal transmission rates for the rotavirus submodel (3.1) and norovirus submodel (3.2).

Fig. 5 compares the forms of the reproductive numbers for rotavirus and norovirus under low (Fig. 5a and 5c) and high (Fig. 5b and 5d) seasonality. Since there is no exposed class in the rotavirus sub-model, the periodic ${\tilde{\mathcal{R}}}_0^R$ and ${\mathcal{R}}_0^R$ under constant transmission are identical, both denoted simply as just ${\mathcal{R}}_0^R$ in Fig. 5a and b. In the case when the amplitude of seasonality is low, the blue curve displays a temperate sinusoidal structure with peaks and dips closer to 8 an 4, respectively (Fig. 5a). When the amplitude of seasonality is high, there is a noticeable variation in seasonal oscillations (Fig. 5b). The blue curve undergoes a significant sinusoidal pattern with clear peaks and dips and portrays a strong dependence on seasonal components. In both cases, the instantaneous ${\hat{\mathcal{R}}}_0^R(t)$ accentuates how the existing transmission rate can differ over time due to seasonal dominance, while constant ${\mathcal{R}}_0^R$ being the average value of transmission rate. Fig. 5c and d illustrates the comparison between instantaneous ${\hat{\mathcal{R}}}_0^N(t)$, periodic ${\tilde{\mathcal{R}}}_0^N$ and reproductive number ${\mathcal{R}}_0^N$ with constant transmission for norovirus. Here, the red dashed line shows a seasonal periodic reproductive number lying between the peaks and dips of the instantaneous ${\hat{\mathcal{R}}}_0^N(t)$. For norovius, there is a difference between the basic reproductive and periodic reproductive numbers; however, this basic reproductive number (black dotted line) lies close to the periodic ${\tilde{\mathcal{R}}}_0^N$ (red dashed line), showing only slight variations. These differences are larger under higher seasonality, Fig. 5d.

Fig. 5.

Effects of strength of seasonality on reproductive numbers (Instantaneous ${\hat{\mathcal{R}}}_0^R(t)$, basic reproductive number ${\mathcal{R}}_0^R$) for rotavirus (a,b) and reproductive numbers (instantaneous ${\hat{\mathcal{R}}}_0^N(t)$, periodic ${\tilde{\mathcal{R}}}_0^N$, basic reproductive number ${\mathcal{R}}_0^N$) for norovirus (c,d). Showing scenarios for low seasonal amplitude (a,c) and high seasonal amplitude (b,d).

Model solutions for the rotavirus submodel depicted in Fig. 6 highlight how infected cases I_R vary over time for different mean transmission rates (β_R0) under different (ϵ₁) values. Here, the middle column shows high transient peaks in the number of infected cases that dampen down to stable periodic solutions over a few decades. The mean and maximum of these long-term stable periodic solutions depends on β_R0 and ϵ₁. Interestingly, for intermediate-low values of β_R0 the amplitudes of the oscillations in the long-term dynamics of the number of I_R cases is largest, especially under high seasonal conditions. For low seasonality, the difference between the maximum and mean I_R cases is small (Fig. 6c). For intermediate seasonality, the peak in oscillatory solutions of I_R diverges from the mean at intermediate-low values of β_R0, close to conditions when ${\mathcal{R}}_0^R\approx 1$ (Fig. 6f). This difference between the mean and maximum number of infected case I_R near conditions when ${\mathcal{R}}_0^R\approx 1$ is largest under high seasonality (Fig. 6i).

Fig. 6.

Effects of varying the seasonal amplitude ϵ₁ for the rotavirus submodel (3.1). Seasonal transmission rates (first column) and infected cases I_R (middle column) over time for three values of mean transmission rate β_R0. Long-term dynamics (after 30 years) of the maximum and mean of the seasonal infected cases {I_R} (last column) for varying mean transmission rates. Low seasonality ϵ₁ = 0.05 (abc), medium seasonality ϵ₁ = 0.1 (def), and high seasonality ϵ₁ = 0.2 (ghi).

While, the average number of infected cases consistently increases with increasing mean transmission rates (red dashed curves in Fig. 6c, 6f, 6i), the seasonal peaks in the maximum number of infected cases (blue curves) observed for relatively low values of mean transmission rates represent an unintuitive behavior.

In a similar manner, Fig. 7 depicts how the transmission rate β_N and the number of norovirus infected individuals I_N grow over time for different mean transmission rates (β_N0) under different (ϵ₂) values. Similar to the rotavirus only submodel, here the mean and maximum of long-term stable periodic solutions for I_N depends on β_N0 and ϵ₂. The differences between the mean and maximum number of infected cases in the long-term exhibits similar dynamics at the rotaviarus model (Fig. 7c, 7f, 7i); however, seasonality drives larger differences between the two for norovirus.

Fig. 7.

Effects of varying the seasonal amplitude ϵ₂ for the norovirus submodel (3.2). Seasonal transmission rates (first column) and infected cases I_N (middle column) over time for three values of mean transmission rate β_N0. Long-term dynamics (after 30 years) of the maximum and mean of the seasonal infected cases {I_N} (last column) for varying mean transmission rates. Low seasonality ϵ₂ = 0.05 (abc), medium seasonality ϵ₂ = 0.1 (def), and high seasonality ϵ₂ = 0.2 (ghi).

Fig. 6, Fig. 7 highlight the phenomenon that the seasonal maximum infected cases peak for unintuitive low values of mean transmission rates in the submodels with a single disease, especially under high seasonal conditions. This phenomenon is also observed in our co-infection model and corresponds with the small yellow region depicted in Fig. 4g, just above the white line threshold where ${\mathcal{R}}_0^R=1$ and in Fig. 4h, just to the right of the white line threshold where ${\mathcal{R}}_0^N=1$. Models that neglect seasonal transmission may be underestimating potential disease burden.

Fig. 6i not only highlights that seasonality increases burden but also implies the dynamics shift from low -level endemicity to periodic resurgences of disease even under the influence of the same mean transmission rates. This observation emphasizes a phase where the risk of outbreak is amplified in a disproportionate manner. In Fig. 7i, the maximum of I_N exceeds 40, while the mean remains below ten, resulting in a ratio greater than four. This transition in behaviour from smooth to sharp outbreaks points out a critical point in seasonality strength that was not evident in lower amplitudes.

5.2.2. Shifts in seasonality and its effects

Here, we explore the shifts of seasonality as variations in the timing of the peaks seasonal transmission rates for the co-infection model (4.1). To explore the effects of shifting the seasonal transmission rates for rotavirus and norovirus relative to each other, we keep the seasonal transmission of rotavirus Eq. (10) and vary that of norovirus by shifting the q in Eq. (11). Fig. 8 reveals how shifting the transmission rates β_R(t) and β_N(t) controls the alignment of their phases. With no shift, when q = 0 for norovirus, β_R(t) and β_N(t) completely aligns with each other indicating their peaks occur simultaneously (Fig. 8a). A quarter period shift i.e. q = 365/4 gives rise to a slight misalignment between the peaks, showing a moderate phase delay between the peaks (Fig. 8b). Finally, when q = 365/2, β_R(t) and β_N(t) are completely out of phase, indicating a de-synchronized pattern (Fig. 8c).

Fig. 8.

Effect of peak shift (q) on the alignment of transmission rates, β_R(t) and β_N(t) under different seasonal conditions.

The impact of the shift q on the magnitude of co-infection is shown in (Fig. 9). As the shift q increases from 0 to 365/2 days from being perfectly aligned to reducing the overlap, the annual cumulative co-infected cases starts to decline reducing to lower combined co-infected cases annually (Fig. 9a). We observe a sudden increase in the burden of co-infection around 35 days and a dramatic fall around 365/2 days, which we identify as thresholds driven by the shits in alignment of seasonal transmission rates (Fig. 9a). The long-term co-infected cases exhibits fluctuations, which implies that the number of co-infections attains a maximum but it also maintains a mean value. Beyond 150 days, the long-term co-infected cases exhibits fluctuations, but largely remain unaltered. This trend indicates a saturation point where the long-term cases decreases as the two seasonal transmission rate become out of sync. The differences between the maximum and mean number of co-infected cases is significant as we shift the alignment of transmission rates Fig. 9b. Here, we unexpectedly get maximum number of cases that range from three to over six times the mean number of cases as we shift the peak. It demonstrates that these phase shifts complicate the dynamics and make it challenging to accurately predict the burden of co-infections. This pattern also indicates that increasing peak shifts q leads to the reduced long-term cases, but not in a continuous sequence due to the misalignment of transmission rates, β_R(t) and β_N(t). Fig. 6, Fig. 7, Fig. 8, Fig. 9 convey that seasonal intensity and the timing of the viruses play an essential roles in co-infection dynamics.

Fig. 9.

(a) Annual cumulative co-infected cases vs peak shift q and (b) long-term cumulative co-infected cases vs peak shift q.

5.3. Global parameter sensitivity analysis

While we conducted sensitivity analyses on the basic reproductive numbers for the submodels of just rotavirus (section 3.1.1) and just norovirus (section 3.2.1), our full co-infected model with seasonality is more complex and has many more parameters. A global sensitivity analysis is more suitable for providing a comprehensive assessment. Here, we conduct a global parameter sensitivity analysis using Latin hypercube sampling (LHS) along with the partial rank correlation coefficient (PRCC) statistical measure, following (Marino et al., 2008). Baseline parameter values and their ranges used in the LHS are shown in Table 1. We consider long-term dynamics of the maximum and mean number of co-infected cases as output measures for the analysis. Generally, parameters had monotonic relationships with these output measures, see Appendix A.3, so the PRCC is an appropriate statistical measure to consider (Marino et al., 2008). Following Marino et al., 2008 (Marino et al., 2008), we performed a z-test on transformed PRCC values to identify significant parameters based on their relative sensitivity. The result of the z-test showed that, in general, parameters with PRCC values of greater magnitude exerted a strong influence on the output measures. The resulting PRCC values are shown in Fig. 10. As expected, the average transmission rates β_R0 and β_N0 are highly influential and positively affect the mean number of co-infected cases, while death (τ_N, τ_R) and recovery (γ_N, γ_R, γ_RN) rates are also influential but negatively affect the mean, Fig. 10a. Additionally, the rate at which recovered individuals lose their temporary immunity δ_RN plays a significant role. The difference between the PRCC results for the maximum and the mean number of co-infected cases is interesting. Here, we see the strengths of seasonality ϵ_R and ϵ_N are more influential on the maximum of I_RN than the mean transmission rates β_R0 and β_N0, Fig. 10b.

Fig. 10.

Sensitivity analysis showing partial rank correlation coefficient (PRCC) for long term dynamics of the (a) mean and (b) maximum number of co-infected cases I_RN. PRCC values range from −1 to 1, where negative values indicate an inverse relationship between the parameter and the output measure, and positive values suggest a direct positive impact on the output. Values highest in magnitude are the most influential. Values marked n.s. are non-significant (statistical significance p > 0.05).

6. Conclusion

This study presents deterministic models for the co-infection of rotavirus and norovirus. We develop SIR models in a continuous-time setting to analyze the prevalence of these diseases and predict the burden of co-infection. Core modeling approaches were discussed using time series plots to understand the dynamics of the diseases and their co-infection. This analysis shows that transmission rates greatly influence the number of infected cases. We extend our model to incorporate seasonality and assess its effects at different times of the year. Both rotavirus and norovirus demonstrate clear seasonal peaks clustered in winter months on an annual basis in temperate settings (Ahmed et al., 2013), (Patel et al., 2013). Such annually symmetric trends in their outbreaks motivate the use of simple sinusoidal forcing functions to capture the seasonal dynamics of their transmission. However, alternative shapes (e.g., asymmetric or pulse-like functions) (Hridoy, 2024) may offer more detailed insights, and complement our conclusions which use seasonal forms with single dominant peaks. This may be particularly important for rotavirus, as seasonal trends have been shown to vary since the introduction of vaccines in 2006 (Tate et al., 2009). We leave more complex seasonal transmission functions and the incorporation of rotavirus vaccines for future work. This research shows how the amplitude of seasonality significantly impacts basic and periodic reproductive numbers. It also provides insights into how the long-term disease dynamics of rotavirus and norovirus are influenced by seasonal forcing. This model also calculates the effects of changes in the peak shifts of the two viruses. Moreover, if the peak shifts in rotavirus and norovirus do not align, it results in varying number of infected individuals. The symmetry and asymmetry of the alignment of the peak shifts influences both maximum and cumulative co-infection levels.

While the presented model offers meaningful insights into the dynamics of rotavirus and norovirus, as well as their co-infection, it neglects to include the vaccinations (which have been developed for rotativus (Dennehy, 2008)), other preventive measures, and behavioral and social interventions. Including these factors could provide more knowledge and enhance the model insights. Alternative investigations could examine the influence of seasonality by varying the timing of infection, specifically by modifying the initial conditions related to infection onset. Given that the stochastic nature of human behavior can influence contact and transmission rates, developing stochastic processes based on our co-infection model framework (4.1), i.e., Continuous -Time Markov Chains, may provide vital insights for better understanding and predicting co-infection dynamics.

CRediT authorship contribution statement

Mohammadi Qurratul Ain: Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Formal analysis, Conceptualization. Angela Peace: Writing – review & editing, Writing – original draft, Visualization, Supervision, Software, Methodology, Formal analysis, Conceptualization.

Declaration of competing interests

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.

Handling Editor: Dr Daihai He

Appendix A. Local stability Analysis

Appendix A.1. Rotavirus sub-model

Considering only the rotavirus infections from the co-infection model, the sub-model of rotavirus is:

$$\begin{array}{l}\frac{dS}{dt}=\lambda -\frac{{\beta }_{R}S{I}_R}{N}-\nu S+{\delta }_R{R}_R\\ \frac{d{I}_R}{dt}=\frac{{\beta }_{R}S{I}_R}{N}-\nu I_R-{\tau }_R{I}_R-{\gamma }_R{I}_R\\ \frac{d{R}_R}{dt}={\gamma }_R{I}_R-\nu R_R-{\delta }_R{R}_R\end{array}$$

Proof of Theorem 3.1.

Proof. The disease-free equilibrium point, denoted by $X_0^R$ is given as $X_0^R=(\frac{\lambda }{\nu },0,0)$.

It is very important to study the basic reproductive number for the detailed study of such models. For evaluating the reproductive number, we use

$$\frac{d{I}_R}{dt}=\frac{{\beta }_{R}S{I}_R}{N}-\nu I_R-{\tau }_R{I}_R-{\gamma }_R{I}_R$$

Similar to the approach used earlier for the NGM, here $F=\frac{{\beta }_{R}S{I}_R}{N}$ and V = νI_R + τ_RI_R + γ_RI_R. For the disease-free scenario, N = S. Hence F′ = β_R and V′ = ν + τ_R + γ_R and ${(V^{\prime })}^{-1}=\frac{1}{\nu +{\tau }_R+{\gamma }_R}$. The basic reproductive number, ${\mathcal{R}}_0$ is defined as ${\mathcal{R}}_0=\rho (F^{\prime }{(V^{\prime })}^{-1})$ which is given by

$${\mathcal{R}}_0^R=\frac{{\beta }_R}{\nu +{\tau }_R+{\gamma }_R}$$ (A.1)

This can also be seen when examining the local stability of the steady state solutions using the linearization method. The Jacobian matrix of the sub-model at $E_0^R$ is given by

$$J(X_0^R)=\left[\begin{array}{lll}-\nu & -{\beta }_R& {\delta }_R\\ 0& {\beta }_R-\nu -{\tau }_R-{\gamma }_R& 0\\ 0& {\gamma }_R& -\nu -{\delta }_R\end{array}\right]$$

Hence, the eigenvalues for the $J(X_0^R)$ are λ₁ = −ν − δ_R, λ₂ = −ν and λ₃ = β_R − ν − τ_R − γ_R. If ${\mathcal{R}}_0^R>1$, then λ₃ > 0. That is, the disease-free equilibrium becomes unstable.

Proof of Theorem 3.2.

Proof. The endemic equilibrium point denoted by $X_E^R$ is given as $X_E^R=(S_E^R,I_E^R,R_E^R)$, where

$$S_E^R=-\frac{\lambda ({\gamma }_R+{\delta }_R+\nu )({\gamma }_R+\nu +{\tau }_R)}{({\tau }_R-{\beta }_R){\nu }^2+({\tau }_R+{\delta }_R+{\gamma }_R)({\tau }_R-{\beta }_R)+{\tau }_R{\delta }_R({\tau }_R-{\beta }_R-{\gamma }_R)}$$ (A.2)

$$I_E^R=\frac{S_E^R(\nu +{\delta }_R)}{{\gamma }_R+{\delta }_R+\nu }\left({\mathcal{R}}_0^R-1\right)$$ (A.3)

$$R_E^R=\frac{{\gamma }_R{I}_E^R}{\nu +{\delta }_R}.$$ (A.4)

First, we will show that $S_E^R$ is nonnegative for positive initial conditions. Suppose $S_E^R(0)>0$, but there is some time t₁ > 0 where $S_E^R(t_1)=0$. Then

$${\left(\frac{d{S}_E^R}{dt}\right|}_{t=t_1}=\lambda +\delta R(t_1)\ge 0$$

and $S_E^R(t)$ is increasing at t = t₁ and will not become negative. Note that R(t₁) ≥ 0 here, as it can also be shown that R(t) remains nonnegative. It is clear that $I_E^R$ is only positive when ${\mathcal{R}}_0^R>1$. $R_E^R$ can be written in terms of $I_E^R$ and is similarly positive when ${\mathcal{R}}_0^R>1$. We note that this is the same condition where the disease-free equilibria is unstable.

The stability of the endemic equilibrium is shown numerically in figure A.11.

Fig. A.11.

Numerical solutions of the rotavirus submodel showing the endemic equilibrium (left) gains stability when ${\mathcal{R}}_0^R>1$. This corresponds with prevalence (right) becoming positive as ${\mathcal{R}}_0^R$ crosses 1. Values of ${\mathcal{R}}_0^R$ varied by varying values of β_R ∈ (0, 1).

Appendix A.2. Norovirus sub-model

If we consider only the presence of norovirus disease, the system is reduced to the following set of equations:

$$\begin{array}{l}\frac{dS}{dt}=\lambda -\frac{{\beta }_{N}S{I}_N}{N}-\nu S+{\delta }_N{R}_N\\ \frac{d{E}_N}{dt}=\frac{{\beta }_{N}S{I}_N}{N}-\nu E_N-{\theta }_N{E}_N\\ \frac{d{I}_N}{dt}={\theta }_N{E}_N-\nu I_N-{\tau }_N{I}_N-{\gamma }_N{I}_N\\ \frac{d{R}_N}{dt}={\gamma }_N{I}_N-\nu R_N-{\delta }_N{R}_N\end{array}$$

Proof of Theorem 3.3.

Proof. The disease-free equilibrium for only norovirus sub-model is given by $X_0^N=(\frac{\lambda }{\nu },0,0,0)$ and the corresponding basic reproductive number is computed as

$${\mathcal{R}}_0^N=\frac{{\beta }_N{\theta }_N}{(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)}$$ (A.5)

Furthermore, we check the local stability of the disease-free equilibrium using the Jacobian matrix.

The Jacobian matrix at $X_0^N$ is

$$J\left(X_0^N\right)=\left[\begin{array}{llll}-\nu & 0& -{\beta }_N& {\delta }_N\\ 0& -\nu -{\theta }_N& {\beta }_N& 0\\ 0& {\theta }_N& -\nu -{\tau }_N-{\gamma }_N& 0\\ 0& 0& {\gamma }_N& -\nu -{\delta }_N\end{array}\right]$$

The eigenvalues for the $J(X_0^N)$ are λ₁ = −ν − δ_N, λ₂ = −ν,${\lambda }_3=-\frac{{\gamma }_N}{2}-\nu -\frac{{\tau }_N}{2}-\frac{{\theta }_N}{2}-\frac{\sqrt{4{\beta }_N{\theta }_N+{\gamma }_N^2+2{\gamma }_N{\tau }_N-2{\gamma }_N{\theta }_N+{\tau }_N^2-2{\tau }_N{\theta }_N+{\theta }_N^2}}{2}and$

${\lambda }_4=-\frac{{\gamma }_N}{2}-\nu -\frac{{\tau }_N}{2}-\frac{{\theta }_N}{2}+\frac{\sqrt{4{\beta }_N{\theta }_N+{\gamma }_N^2+2{\gamma }_N{\tau }_N-2{\gamma }_N{\theta }_N+{\tau }_N^2-2{\tau }_N{\theta }_N+{\theta }_N^2}}{2}$

If λ₃ > 0 then.$-\frac{{\gamma }_N}{2}-\nu -\frac{{\tau }_N}{2}-\frac{{\theta }_N}{2}-\frac{\sqrt{4{\beta }_N{\theta }_N+{\gamma }_N^2+2{\gamma }_N{\tau }_N-2{\gamma }_N{\theta }_N+{\tau }_N^2-2{\tau }_N{\theta }_N+{\theta }_N^2}}{2}>0$

$$\begin{array}{l}\frac{\sqrt{4{\beta }_N{\theta }_N+{\gamma }_N^2+2{\gamma }_N{\tau }_N-2{\gamma }_N{\theta }_N+{\tau }_N^2-2{\tau }_N{\theta }_N+{\theta }_N^2}}{2}>\frac{{\gamma }_N}{2}+\nu +\frac{{\tau }_N}{2}+\frac{{\theta }_N}{2}\\ \sqrt{4{\beta }_N{\theta }_N+{\gamma }_N^2+2{\gamma }_N{\tau }_N-2{\gamma }_N{\theta }_N+{\tau }_N^2-2{\tau }_N{\theta }_N+{\theta }_N^2}>{\gamma }_N+{\tau }_N+{\theta }_N+2\nu \\ 4{\beta }_N{\theta }_N+{\gamma }_N^2+2{\gamma }_N{\tau }_N-2{\gamma }_N{\theta }_N+{\tau }_N^2-2{\tau }_N{\theta }_N+{\theta }_N^2>{({\gamma }_N+{\tau }_N+{\theta }_N+2\nu )}^2\\ 4{\beta }_N{\theta }_N+{\gamma }_N^2+2{\gamma }_N{\tau }_N-2{\gamma }_N{\theta }_N+{\tau }_N^2-2{\tau }_N{\theta }_N+{\theta }_N^2>{({\gamma }_N+{\tau }_N+{\theta }_N)}^2+{\left(2v\right)}^2\end{array}$$

which can be written as

$$\begin{array}{l}4{\beta }_N{\theta }_N>4{\tau }_N{\theta }_N+4{\gamma }_N{\theta }_N+4{\nu }^2+4\nu {\gamma }_N+4\nu {\tau }_N+4\nu {\theta }_N\\ {\beta }_N{\theta }_N>{\tau }_N{\theta }_N+{\gamma }_N{\theta }_N+{\nu }^2+\nu {\gamma }_N+\nu {\tau }_N+\nu {\theta }_N\\ {\beta }_N{\theta }_N>(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)\\ \frac{{\beta }_N{\theta }_N}{(\nu +{\theta }_N)(\nu +{\tau }_N+{\gamma }_N)}>1\\ {\mathcal{R}}_0^N>1\end{array}$$

That is, the disease-free equilibrium becomes unstable.

Proof of Theorem 3.4.

Proof. The endemic equilibrium point denoted by $X_E^N$ is given as $X_E^N=(S_E^N,I_E^N,R_E^N)$, where

$$S_E^N=-\frac{\lambda \left(\nu +{\tau }_N+{\gamma }_N\right)\left(\nu +{\theta }_N\right)A}{\left(\left({\tau }_N-{\beta }_N\right){\nu }^3+\left({\tau }_N+{\delta }_N+{\gamma }_N+{\theta }_N\right)\left({\tau }_N-{\beta }_N\right){\nu }^2+B\right){\theta }_N}$$ (A.6)

$$E_E^N=\frac{S_E^N\left({\mathcal{R}}_0^N-1\right)\left(\nu +{\tau }_N+{\gamma }_N\right)\left({\delta }_N+\nu \right)}{\left({\nu }^2+\left({\tau }_N+{\delta }_N+{\gamma }_N+{\theta }_N\right)\nu +{\delta }_N{\tau }_N+\left({\delta }_N+{\gamma }_{NN}\right){\theta }_N+{\delta }_N{\gamma }_N\right)}$$ (A.7)

$$I_E^N=\frac{{\theta }_N{E}_E^N}{\nu +{\tau }_N+{\gamma }_N}$$ (A.8)

$$R_E^N=\frac{{\gamma }_N{I}_E^N}{\nu +{\delta }_N}.$$ (A.9)

where

$$A=\left({\nu }^2+\left({\tau }_N+{\delta }_N+{\gamma }_N+{\theta }_N\right)\nu +\left({\delta }_N+{\gamma }_N\right){\theta }_N+{\delta }_N\left({\tau }_N+{\gamma }_N\right)\right)$$

and

$$B=\left(\left({\tau }_N+{\delta }_N+{\gamma }_N\right){\theta }_N+{\delta }_N\left({\tau }_N+{\gamma }_N\right)\right)\left({\tau }_N-{\beta }_N\right)\nu +{\tau }_N{\delta }_N{\theta }_N\left({\tau }_N-{\beta }_N+{\gamma }_N\right)$$

Similar to the proof in theorem 3.2, we can see that $S_E^N$ is nonnegative for positive initial conditions. It is clear that $E_E^N$ is only positive when ${\mathcal{R}}_0^N>1$. $I_E^N$ and $R_E^N$ can be written in terms of $E_E^N$ and are similarly positive when ${\mathcal{R}}_0^N>1$. We note that this is the same condition where the disease-free equilibria is unstable.

The stability of the endemic equilibrium is shown numerically in figure A.12.

Fig. A.12.

Numerical solutions of the norovirus submodel showing the endemic equilibrium (left) gains stability when ${\mathcal{R}}_0^N>1$. This corresponds with prevalence (right) becoming positive as ${\mathcal{R}}_0^N$ crosses 1. Values of ${\mathcal{R}}_0^N>1$ varied by varying values of β_N ∈ (0, 3).

Appendix A.3. Parameter sensitivity analysis monotonicity results

The monotonic relationships between the parameters and output measures are shown in Figs. (A.13, A.14). Here, we varied parameters using Latin Hypercube sample (LHS) one at a time to see the relationship between each parameter and the output measures for the mean and maximum number of co-infected cases, I_RN. While the majority of the dynamics are monotonic, there are some non-monotonic trends. However, the variation in the output measures in these non-monotonic regions are minimal. Therefore, we decided to use the Partial Rank Correlation Coefficient (PRCC) technique to conduct a global sensitivity analysis and identify key parameters.

Fig. A.13.

Individual parameter variations from the Latin hypercube Sampling (LHS) showcasing general monotonic trends between parameters and the output measure for long-term dynamics of the mean number of co-infected individuals, I_RN.

Fig. A.14.

Individual parameter variations from the Latin hypercube Sampling (LHS) showcasing general monotonic trends between parameters and the output measure for long-term dynamics of the maximum number of co-infected individuals, I_RN.