Modeling the Spread of Methicillin-Resistant Staphylococcus aureus in Nursing Homes for Elderly

Methicillin-resistant Staphylococcus aureus (MRSA) is endemic in many hospital settings, including nursing homes. It is an important nosocomial pathogen that causes mortality and an economic burden to patients, hospitals, and the community. The epidemiology of the bacteria in nursing homes is both hospital- and community-like. Transmission occurs via hands of health care workers (HCWs) and direct contacts among residents during social activities. In this work, mathematical modeling in both deterministic and stochastic frameworks is used to study dissemination of MRSA among residents and HCWs, persistence and prevalence of MRSA in a population, and possible means of controlling the spread of this pathogen in nursing homes. The model predicts that: without strict screening and decolonization of colonized individuals at admission, MRSA may persist; decolonization of colonized residents, improving hand hygiene in both residents and HCWs, reducing the duration of contamination of HCWs, and decreasing the resident∶staff ratio are possible control strategies; the mean time that a resident remains susceptible since admission may be prolonged by screening and decolonization treatment in colonized individuals; in the stochastic framework, the total number of colonized residents varies and may increase when the admission of colonized residents, the duration of colonization, the average number of contacts among residents, or the average number of contacts that each resident requires from HCWs increases; an introduction of a colonized individual into an MRSA-free nursing home has a much higher probability of leading to a major outbreak taking off than an introduction of a contaminated HCW.

By setting the RHS of (1) in the main text equaling zeros, the disease-present steady state is as follows: where P (C * ) is a polynomial function of C * and C * satisfies the following equation: It is clear that c 0 is always negative and c 3 is always positive. We study the existence and uniqueness of the disease-present steady state by considering roots of P (C * ) or using Descarte's rule of signs. We consider four cases of λ and β r according to whether there are admission of colonized residents and contacts among residents.
1. λ > 0 and β r > 0. By further assuming that β h > β r , it is obvious that c 2 is positive for all positive values of parameters (as (1 − λ)α h N r γ c > 0). We rewrite the polynomial for C * as follows: Either c 1 is positive or negative, there is only one sign change in P (C * ) (+,+,+,-, or +,+,-,-). By Descarte's rule, there is one positive real root of P (C * ). Set κ = −C * . Hence, Whether c 1 is positive or negative, there are two sign changes in P (κ) (+,-,+,+, or +,-,-,+). By Descarte's rule, there are either two or zero positive real roots of P (κ) or in other words two or zero negative real roots of P (C * ). Therefore, there are exactly one positive real root and possibly two negative real roots or complex conjugate roots of P (C * ). In case β h > β r is not assumed, it is possible that c 2 can be either positive or negative. If it is positive, it follows from the above result we have just shown. If not, let c 2 = −C 2 . Hence, Substituting this argument into c 1 , we obtain Consequently, if c 2 is negative, we find that c 1 is also negative. By considering the change of signs, there is only one sign change (+,-,-,-) in P (C * ) and there are two sign changes in P (κ). Hence, there are only one positive real root and possibly two negative real roots or complex conjugate roots of P (C * ). In conclusion, there always exists a unique non-zero positive real root C * for all λ > 0 and β r > 0.
2. λ > 0 and β r = 0. This is likely to occur in nursery homes with bed-bound residents or private houses. We have c 3 = 0 in this case. Clearly, the coefficient c 2 is positive. We do not have information about the sign of c 1 . Hence, we consider both positive and negative values of c 1 . We write the polynomial P (C * ) as follows: The explicit real solutions of P (C * ) are Because C 1 < √ C 2 1 + 4C 0 C 2 , we can always find a positive real root of P (C * ) such that Hence, there exists a unique non-zero positive real root when λ > 0 and β r = 0.
3. λ = 0 and β r > 0. We rewrite the coefficient c 1 as follows: The polynomial P (C * ) becomes Obviously, zero is one of the roots of P (C * ). In case c 2 is negative, we can easily show that c 1 is also negative (see above). Consequently, there exists a unique non-zero positive real root of P (C * ) which is If c 2 is positive, c 1 can be either positive or negative. However, if c 1 is positive, the other two roots of P (C * ) are either both real and negative or complex conjugates. There exists a unique non-zero positive real root of P (C * ) if only if c 1 is negative or R 0 > 1. Hence, a sufficient condition for a unique non-zero real positive root of P (C * ) is R 0 > 1.
4. λ = 0 and β r = 0. The coefficient c 2 is always positive and we write The polynomial P (C * ) can be written in the following form: Hence, there exists a unique non-zero positive real root In conclusion, for all λ > 0 and β r ≥ 0, there always exists a disease-present steady state. In case λ = 0, for all β r ≥ 0 there exists the disease-present steady state if and only if R 0 > 1.

Stability analysis
We first study long-term dynamics of (1) when there is admission of colonized residents (λ > 0). In this case, there only exists a disease-present steady state. The Jacobian matrix of (1) at the disease-present steady state is The characteristic equation of J * is At the disease-present steady state (from the right-hand sides of (1) equaling zeros), we have Substituting these two terms into a 1 and a 2 , we obtain Clearly, a 1 > 0 and a 2 > 0. By the Routh-Hurwitz criteria, the disease-present steady state is stable for all positive C * . Because we can always find a unique non-zero real positive root of P (C * ) for all λ > 0 and β r ≥ 0, the disease-present steady state is stable. Secondly, let us consider a stability condition of (1) when there is no admission of colonized residents (λ = 0). When λ = 0, there are two steady states: the disease-free and present steady states. The disease-free steady state always exists. However, the disease-present steady state exists if and only if R 0 > 1. At the disease-free steady state, the Jacobian matrix is Consequently, we have , and det(J 0 ) = µ(ω + γ c )(1 − R 0 ).
If R 0 < 1, we have det(J 0 ) > 0, and trace(J 0 ) < 0 because βr (ω+γc+µ) < R 0r < 1. By the Routh-Hurwitz criteria, the disease-free steady state is stable if R 0 < 1. In a similar way in deriving a stability condition for the disease-present steady state when λ > 0, the disease-present steady state is stable if R 0 > 1. Therefore, the disease-free steady state is stable if and only if R 0 < 1 and the disease-present steady state is stable if and only if R 0 > 1.

Hand hygiene compliance
One of the methods to investigate hand hygiene compliance in mathematical models is by including the term (1 − η), where η is the fraction of HCW/resident hand hygiene compliance (η = 0 means no compliance and η = 1 means perfect compliance) in the transmission terms (DAgata et al. [1]). This (1 − η) term consequently reduces the transmission rate or the probability of colonization when the number of contacts is fixed. However, we do not consider this factor in detail but only demonstrate that hand hygiene compliance that may relate to the probability of colonization in residents may help to reduce MRSA prevalence in nursing homes. Figure 1A shows that the prevalence of MRSA increases when the probability of colonization in residents by contacting with colonized residents and the number of contacts among them increase. It also increases when the probability of colonization in residents by contacting with contaminated HCWs and the number of contacts between residents and HCWs increase.

Derivation of the mean, variance, and covariance equations of the stochastic model
We study a stochastic model based on the ODE system (1) by using the continuous time Markov chain process (CTMC). In the model, time is continuous but state variables are discrete. Because we assume that both of the total populations (residents and HCWs) are constant, the process is bivariate, A joint probability function is given by This bivariate process has the Markov property and the transition probabilities are shown in Table 2.
From transition probabilities, we can write the forward Kolmogorov differential equations as follows: dp(C,Hc) dt These equations can be used to derive formulae for the rates of change of the expected numbers of colonized residents and contaminated HCWs (E(C), E(H c )), and the higher moments such as variances and covariance. Here, we introduce the moment generating function (MGF). Define We can rewrite the forward Kolmogorov equations in terms of the moment generating function as follows: To derive the equations for means, variances, and covariance of the stochastic model, it is more convenient to use the cumulant generating function which is a logarithm of the moment generating function. Define The time derivative for the cumulant generating function K is given by ) .
The cumulant generating function can be expanded in terms of the cumulants k ij as follows: By substituting this power series into the time derivative equation of K, the time evolution of the moments of orders one and two is described by: