Control Strategies in Multigroup Models: The Case of the Star Network Topology

Bulletin of Mathematical Biology manuscript No. (will be inserted by the editor) Control Strategies in Multigroup Models: The Case of the Star Network Topology Fernando Salda˜na ·Ignacio Barradas Received: date / Accepted: date Abstract In this paper, we propose control strategies for multigroup epidemic models. We use compartmental SIRS models to study the dynamics of nhost groups sharing the same source of infection in addition to the transmission among members of the same group. In particular, we consider a model for infectious diseases with free-living pathogens in the environment and a metapopulation model with a central patch. We give the detailed derivation of the target reproduction number under three public health interventions and provide the corresponding biological insights. Moreover, using the nextgeneration approach, we calculate the basic reproduction numbers associated with subsystems of our models and determine algebraic connections to the target reproduction number of the complete model. The analysis presented here illustrates that understanding the topological structure of the infection process and partitioning it in simple cycles is useful to design and evaluate control strategies. Keywords Control strategies ·Target reproduction number ·Multigroup model ·Star Network ·Basic reproduction number. 1 Introduction An omnipresent quantity in the modeling of infectious diseases is the basic reproduction number R0(Heesterbeek,2002). The value of R0not only is an indicator of the severity of an epidemic, it is also a powerful tool to estimate Fernando Salda˜na E-mail: [email protected] Ignacio Barradas E-mail: [email protected] Centro de Investigaci´on en Matem´aticas, Apartado Postal 402, C.P. 36240, Guanajuato M´exico 2 Fernando Salda˜na, Ignacio Barradas the control effort needed to eradicate a disease when treating homogeneous populations. In fact, for homogeneous populations, it is known (Keeling et al., 2013) that an infection can be eliminated provided that a proportion of individuals greater than 1-1/R0have been afforded lifelong protection. However, for a large number of infectious diseases the host population is not homogeneous (Roberts and Heesterbeek,2003). Furthermore, in real situations, there are multiple constraints (economic and geographical considerations for example) that may significantly affect the design of control policies. Therefore, it is not uncommon that control strategies are not aimed at all host individuals; instead, they treat a particular group of individuals or interactions between them. When designing control measures that target specific types of individuals in heterogeneous populations, the concept of the target reproduction number proposed in (Shuai et al.,2013) arises naturally to quantify the effort needed to control epidemic outbreaks. The basic idea behind the method of Shuai et al. (Shuai et al.,2015) is that it is possible to reduce the value of R0by controlling a specific set Sof entries of the next-generation matrix K= [kij ]. Let KSbe the target matrix corresponding to the target set Sdefined by [KS]ij =kij if (i, j)∈Sand 0, otherwise. The target reproduction number TSfor the target set Sis defined as the spectral radius of the matrix KS·(I−K+KS)−1, provided that ρ(K−KS)<1, where Iis the identity matrix and ρdenotes the spectral radius. This last condition had been called the controllability condition (Knipl,2016), since whenever ρ(K−KS)>1 the disease cannot be eradicated by targeting only S(Shuai et al.,2013). However, since the term controllability evokes formal control theory, we will not use this terminology. Analogously to R0, the target reproduction number has the property that if a proportion bigger than 1−1/TSof the Sentries in Kcan be reduced, then the disease can be eradicated. Moreover, when the next-generation matrix is irreducible then R0<1 if and only if TS<1 for a general target set S, as long as the target reproduction number is well defined. On the other hand, for many diseases the structure of the infection process can be subdivided as a combination of different cycles of infection (Olmos et al.,2015) and it is logical to speculate if there is a relationship between the topological structure of the cycles of infection and the optimal design of public health interventions. Therefore, we propose to examine and control the local dynamics of an infection with a heterogeneous host population to prevent a major outbreak at the global level. In particular, our focus will be models with multiple host types whose infection process follows the star network topology (Edwards et al.,2010) i.e. models with a common source of infection for all the groups but not direct transmission between them. Using these ideas, in this paper we establish connections among the target reproduction number for the whole model and R0’s of sub-models that facilitate the design and evaluation of control strategies. In section 2, we illustrate our method with an epidemiological model for leptospirosis in humans and animals. Different control strategies are proposed and the target reproduction number is computed in each case. General models with nhosts types Control Strategies in Multigroup Models 3 are studied in section 3. In particular, we consider a compartmental model for infections with free-living pathogens in the environment and a metapopulation model with a central patch. In the last section, we summarize and interpret our results. 2 Leptospirosis as an example of an environmentally-driven disease Leptospirosis is a bacterial disease caused by certain members of the genus Leptospira. Except for Antarctica, leptospirosis is present in all continents (Adler and de la Pe˜na Moctezuma,2010) with more than 500,000 cases per year reported globally (Ullmann and Langoni,2011). Although rodents have been recognized as the most important reservoirs of leptospiral infection; humans, domestic pets, cattle and almost every mammal can also contract this infection (Evangelista and Coburn,2010). Environmental conditions play an important role in the transmission of leptospirosis. In fact, leptospires can survive in moist soil and fresh water for a few weeks up to several months (Ko et al.,1999). Moreover, both humans and animals become infected with leptospires through close contact with water, food, or soil contaminated mainly with the urine of reservoir animals (Ganoza et al.,2006). Human to human transmission is also possible by sexual intercourse, transplacentally from mother to fetus and via breast milk to a child. Urine from a patient suffering from leptospirosis should be considered infectious (Terpstra,2003). Several mathematical models have been proposed to study the transmission of leptospirosis in a population over time. In (Holt et al.,2006) the authors proposed a model to study the spread and maintenance of leptospirosis in rodents. One of the first compartmental models for leptospirosis that considered both human and animals was proposed in (Triampo et al.,2007). In their work, they proposed a deterministic SIRS (susceptible-infectious-recoveredsusceptible) model and considered real data of leptospirosis in Thailand. Recently, in (Baca-Carrasco et al.,2015) an epidemiological model that included explicitly a variable for the leptospires in the environment was proposed. This model considers the indirect infections caused by the bacteria present in the environment; however, it did not include possible direct transmissions between humans. Here we present a two-group epidemiological SIRS model for the dynamics of leptospiral infection in humans and animals. Given the importance of the indirect infections caused by contact with a contaminated environment, we consider an equation for the free-living leptospira in the environment as in (Baca-Carrasco et al.,2015) but we also allow direct transmissions among humans. To derive our model, let Si(t), Ii(t) and Ri(t) denote the number of susceptible, infectious and recovered individuals of host type i∈ {1,2} at time t, with i= 1 representing humans and i= 2 representing animals. Therefore the total population size of host type iat time tis given by Ni(t) = 4 Fernando Salda˜na, Ignacio Barradas Si(t) + Ii(t) + Ri(t) for i∈ {1,2}. The variable P(t) represent the amount of leptospira present in the environment. We assume that individuals enter to the population as susceptibles at a constant rate Λi, and die at a per capita death rate µi,i= 1,2. No additional death due to the disease is considered. Susceptible humans can be infected by direct (sexual) contact with other humans at a rate β11 and indirectly via the external environment at a rate β1p. New infections in animals are also due to direct contact with infected animals at a rate β22 and through indirect exposure with a contaminated environment at a rate β2p. Infectious individuals of host type i∈ {1,2}recover at a constant rate γiproportional to the size of their class and the loss of immunity occurs at an average time of 1/νi. The presence of leptospira in the environment increases due to the shedding of infected humans and animals at rates c1and c2, respectively. The bacteria survive in the environment for a mean time of 1/µp. Under these hypotheses, our model is given by the following system of differential equations: ˙ Si=Λi−(βiiIi/Ni+βipP)Si−µiSi+νiRi, ˙ Ii=(βiiIi/Ni+βipP)Si−(µi+γi)Ii, i ∈ {1,2} ˙ Ri=γiIi−(νi+µi)Ri, ˙ P=c1I1+c2I2−µpP. (1) All the parameters are considered nonnegative. Note that we assumed mass action incidence for the indirect (environment-to-host) transmission and standard incidence for the direct (host-to-host) transmission. It is easy to see that the total population size for any host type is asymptotically constant since ˙ Ni=Λi−µiNi, i ∈ {1,2}. The unique disease free equilibrium of system (1) is given by ε0= (N∗ 1,0,0, N∗ 2,0,0,0) (2) with N∗ i=Λi/µifor i∈ {1,2}. Disregarding the infections caused by the contaminated environment it is easy to calculate the basic reproduction number associated with the direct transmission of the disease for any host type, Ri=βii µi+γi , i ∈ {1,2}(3) and Rigives the average number of secondary infectious corresponding to the direct transmission cycle for the ith host-type. The dynamics of the infection process for system (1) is shown in Fig. 1. As many examples in the literature show (Baca-Carrasco and VelascoHern´andez,2016;Chen et al.,2009;Martcheva,2015;Olmos et al.,2015), the Control Strategies in Multigroup Models 5 Fig. 1: Topological structure of the infection process for model (1). In this case, there are two infection cycles, for both, the indirect transmission (R1p and R2p) and the direct transmission (R1and R2). expressions for R0in multigroup models can be very complex, so it would be very useful when evaluating public health interventions to have an expression of R0in terms of R0’s of sub-models of the whole system. This will allow the design of local strategies such as control of specific types of individuals or even interactions between them to control the disease at the global level. As previously mentioned, when targeting specific types of individuals or interactions it is easier to work with the target reproduction number TSthan with R0. Moreover, we know TSand R0are directly related. In fact, if the next-generation matrix K= [kij] is irreducible then R0<1 if and only if TS<1 when the target reproduction number is well defined for the target set S(Shuai et al., 2013,2015). In general terms, to compute the next-generation matrix it is necessary to determine the subsystem that describes the production of new infections and changes in state among infected individuals. The Jacobian matrix Jcorresponding to the linearization of this subsystem at the infection-free equilibrium is decomposed as F−V. The matrix Fis the transmission part, describing the production of new infections, and Vdescribes changes in status, such as recovery or death (Knipl,2016;Martcheva,2015;Van den Driessche and Watmough,2002). However, different interpretations of the disease process can lead to different decompositions into matrices Fand V. Consequently, different nextgeneration matrices can be obtained for a compartmental model. This problem is of particular relevance for infections that can be caused by contact with a contaminated environment since the role of the environment can be interpreted in multiple ways (Bani-Yaghoub et al.,2012). In this work, we assume that the environment acts as a reservoir of the infection. Therefore, the pathogen shedding by infectious hosts are placed in the matrix F. These assumptions give the following matrices Fand Vfor model (1), computed at the disease-free equilibrium (2): F=  β11 0β1pN∗ 1 0β22 β2pN∗ 2 c1c20 and V=  µ1+γ10 0 0µ2+γ20 0 0 µp  6 Fernando Salda˜na, Ignacio Barradas The product FV−1gives the following next-generation matrix K=          R10β1pN∗ 1 µp 0R2 β2pN∗ 2 µp c1 µ1+γ1 c2 µ2+γ2 0          ,(4) which is clearly an irreducible matrix. Using (4) it is not difficult to show that the basic reproduction numbers for the indirect transmission cycle (see figure 1) are given by the following formula Rip =sciβipN∗ i µp(µi+γi), i ∈ {1,2}.(5) Indeed, the characteristic polynomial of the next-generation matrix (4) is P(λ) = λ3−(R1+R2)λ2+ R1R2− 2 X i=1 ciβipN∗ i µp(µi+γi)!λ + (R1+R2) 2 X i=1 ciβipN∗ i µp(µi+γi)− 2 X i=1 ciβipN∗ iRi µp(µi+γi)!. If we set the reproduction numbers associated with the direct transmission of the disease equal to zero, we get ˜ P(λ) = λ3− 2 X i=1 ciβipN∗ i µp(µi+γi)!λ. In order to obtain, for example, R1pwe only need to compute the root of maximum modulus of ˜ P(λ) when the transmission parameters between host type 2 and the contaminated environment are zero i.e. β2p=c2= 0. Hence, Rip satisfies (5). The expression for the R0of the complete model is quite complicated and we are not going to show it here. Instead, we are going to work with the target reproduction number since we know that reducing this quantity below 1 ensures the elimination of the disease, see Theorem 1below. A wide variety of public health interventions can be of help to reduce the spread of leptospirosis. We analyze some of them in the following section. 2.1 Control strategies in the two-group model for leptospirosis Leptospirosis can be considered an environmentally-driven disease because of the important role of environmental conditions in the transmission of the disease. Different control strategies could be applied to reduce the prevalence of environmentally-driven diseases in general and leptospirosis in particular. Control Strategies in Multigroup Models 7 (A) Minimizing exposures to environmental risk factors. The vast majority of human and animal infections with leptospira arise from contact with contaminated water (Terpstra,2003). Therefore, one effective way to minimize the chances of infection is to avoid contact with contaminated water. In humans, factors related to occupational and recreational activities can represent an increased risk of infection. For example, farmers may be exposed to water contaminated by the urine of rodents or other animals when irrigating fields (Terpstra,2003). Hence, where appropriate, protective clothing should be worn to reduce the risk of infection. When preventive measures aim to reduce exposure to environmental risk factors, the target set is S={(1,3),(2,3)}. Following the notation in (Shuai et al.,2013) the target matrix KSis defined by, [KS]ij =kij,if (i, j)∈S 0,otherwise, hence the condition ρ(K−KS)<1 becomes max{R1,R2}<1.(6) This condition is logical since we cannot expect to eradicate a disease by just reducing the indirect infections caused by environmental factors when any of the basic reproduction numbers associated to the host-to-host transmission cycle has a value above 1. Provided that the condition (6) is fulfilled, the target reproduction number is defined as the spectral radius of the matrix KS·(I−K+KS)−1given by          R2 1p 1− R1 c2β1pN∗ 1 µp(µ2+γ2)(1 − R2) β1pN∗ 1 µp c1β2pN∗ 2 µp(µ1+γ1)(1 − R1) R2 2p 1− R2 β2pN∗ 2 µp 0 0 0          . Therefore TS=R2 1p 1− R1 +R2 2p 1− R2 .(7) Hence, TSis an increasing function of each of the local basic reproduction numbers when it is well defined. (B) Reducing the shedding rate. The reduction of certain animal reservoirs such as rodents, immunization of dogs and livestock and prompt treatment after symptoms in humans are all valid ways to reduce the replenishing of pathogen load in the environment due to the excrete of leptospira by infected hosts. 8 Fernando Salda˜na, Ignacio Barradas For control strategies that focus on the reduction of the pathogen shedding rate, the target set is U={(3,1),(3,2)}. Thus, the target matrix is: KU=     0 0 0 0 0 0 c1 µ1+γ1 c2 µ2+γ2 0      . Similar to the case (A), we have ρ(K−KU)<1⇐⇒ max{R1,R2}<1.(8) Moreover, given that (I−K+KU)−1=       1 1− R1 0β1pN∗ 1 µp(1 − R1) 01 1− R2 β2pN∗ 2 µp(1 − R2) 0 0 1        as long as the condition (8) is fulfilled the target reproduction number for the set Uis given by TU=R2 1p 1− R1 +R2 2p 1− R2 ,(9) which coincides with the formula (7). This reflects the fact that there is a reciprocal feedback between strategies (A) and (B) i.e. when the shedding rate is reduced, the risk of getting infected through contact with a contaminated environment also decreases and vice versa. (C) Combination of approaches (A) and (B). In this case the target set is W={(1,3),(2,3),(3,1),(3,2)}. Therefore, the condition ρ(K−KW)<1 is equivalent to max{R1,R2}<1. The characteristic polynomial of the matrix KW·(I−K+KW)−1is P(λ) = −λ3+ R2 1p 1− R1 +R2 2p 1− R2!λ. Therefore, when it is well defined, the target reproduction number is given by TW=sR2 1p 1− R1 +R2 2p 1− R2 .(10) From the formulas for TS,TUand TWit is easy to note that 1<TW<TS=TU,TS=TU<TW<1, or TS=TU=TW= 1 which is not surprising since the intervention strategy defined by the set Wis stronger than the ones defined by Sand U(Shuai et al.,2013). Control Strategies in Multigroup Models 9 2.2 Numerical simulations In this section, we explore numerically model (1). We retrieved some parameters from the literature (Baca-Carrasco et al.,2015;Holt et al.,2006;Leirs et al.,1997) and we gathered them in Table 1. In compartmental epidemic models, a classical procedure to reduce disease spread is to choose a feasible set of model parameters θ={θ1, θ2, . . . , θn} and determine how these parameters should be changed to prevent an epidemic. In our context, this procedure let us to define the target set Sas the set of indices S={(i1, j1),...,(im, jm)}, with the property that if (i, j)∈Sthen the entry kij of Kdepends at least on one of the parameters in θ. On the other hand, if (i, j)/∈Sthen kij is independent of the parameters in θ. If a proportion bigger than 1 −1/TSof the entries Sin Kcan be reduced, then the disease can be eradicated. In particular, replacing the entry kij in Kby kij/TSwhenever (i, j)∈S, a controlled next-generation matrix Kcis formulated [Kc]ij =kij/TS,if (i, j)∈S kij,otherwise Parameter Range Units Λ1[6.0,29.0] person day−1 β11 [1 ×10−3,5×10−2] day−1 β1p[3 ×10−3,6×10−3] leptospires−1day−1 µ1[1/29200,1/14600] day−1 γ1[1/20,1/7] day−1 ν1[1/365,1/10] day−1 Λ2[0.5,4.0] animal day−1 β22 [1 ×10−3,7×10−2] day−1 β2p[1 ×10−3,9×10−3] leptospires−1day−1 µ2[1/10950,1/365] day−1 γ2[1/30,1/10] day−1 ν2[1/90,1/30] day−1 c1[8.4×10−8,2.1×10−7] lestospires person−1day−1 c2[2 ×10−4,2×10−2] lestospires animal−1day−1 µp[1/180,1/4] day−1 Table 1: Estimation of parameters for the two-group model of leptospiral infection in humans and animals (1). 16 Fernando Salda˜na, Ignacio Barradas Furthermore, using induction it can be proved that the characteristic polynomial of the matrix KW·(I−K+KW)−1                 0 0 · · · 0β1pN∗ 1 µp 0 0 · · · 0β2pN∗ 2 µp . . .. . ....0. . . 0 0 · · · 0βnpN∗ n µp c1/(µ1+γ1) (1 − R1) c2/(µ2+γ2) (1 − R2)· · · cn/(µn+γn) (1 − Rn)0                 is given by P(λ) = (−1)n+1λn+1 + (−1)n R2 1p 1− R1 +· · · +R2 np 1− Rn!λn−1.(17) By definition, the value of the target reproduction number TWis equal to the root of largest modulus of the polynomial (17) which it is given by the righthand side of (15). ut Several properties make the target reproduction number such a useful quantity. First, it shares the threshold property of the basic reproduction number (TW<1 if and only if R0<1; and TW= 1 if and only if R0= 1). Second, if the values of the entries Win the next-generation matrix Kare reduced by a proportion more than 1 −1/TW, then the disease will die out. The only requirements for this to be true is that ρ(K−KW)<1 and the next-generation matrix Kto be irreducible. As a decision maker, one might be concerned about how to lower the value of the target reproduction number below 1. In this case, it is important to know how sensitive is TWto the values of the reproduction numbers associated to the environment-to-host and host-to-host transmission cycles to decide on which of the two to invest. This can be done, for instance, using the partial derivatives of TW. For example, from (15) it is easy to prove that, ∂TW ∂Ri ≤∂TW ∂Rip ⇐⇒ Rip 2(1 − Ri)≤1, i = 1,2, . . . , n. Therefore, TWis more sensitive to the reproduction number Rip than to Riif and only if Rip ≤2(1−Ri), see Fig. 3. If other things being equal, this simple analysis tells us if it is more convenient to reduce Rior Rip when controlling the ith host type. However, this sensitivity is local since it does not take into account the simultaneous variation of input parameters. The following theorem establishes the generalization of the result obtained for approach (A) in section 2.1. Control Strategies in Multigroup Models 17 0.0 0.2 0.4 0.6 0.8 1.0  i 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00  ip Fig. 3: Region (blue) in which the target reproduction number TWis more sensitive to the reproduction number associated to the environment-to-host transmission Rip than to the reproduction number of the host-to-host transmission Ri. Theorem 3 Consider the epidemiological model (12) and define the target set as S={(1, n + 1),(2, n + 1),...,(n, n + 1)}. Provided that the condition max{R1,R2,...,Rn}<1 (18) is satisfied, the target reproduction number TSis given by TS=R2 1p 1− R1 +R2 2p 1− R2 +· · · +R2 np 1− Rn .(19) Proof By definition, the target matrix KSis given by KS=             0 0 · · · 0β1pN∗ 1 µp . . .. . .· · · . . .. . . 0 0 · · · 0βnpN∗ n µp 0 0 · · · 0 0             .(20) Consequently, the matrix K−KSis a triangular matrix with R1,...,Rnand 0in its diagonal, hence ρ(K−KS)<1is equivalent to condition (18). The 18 Fernando Salda˜na, Ignacio Barradas product KS·(I−K+KS)−1is given by the following matrix              R2 1p 1− R1 c2β1pN∗ 1 µp(µ2+γ2)(1 − R2) c3β1pN∗ 1 µp(µ3+γ3)(1 − R3) · · · cnβ1pN∗ 1 µp(µn+γn)(1 − Rn) β1pN∗ 1 µp c1β2pN∗ 2 µp(µ1+γ1)(1 − R1) R2 2p 1− R2 c3β2pN∗ 2 µp(µ3+γ3)(1 − R3) · · · cnβ2pN∗ 2 µp(µn+γn)(1 − Rn) β2pN∗ 2 µp . . . . . . . . .· · · . . . . . . c1βnpN∗ n µp(µ1+γ1)(1 − R1) c2βnpN∗ n µp(µ2+γ2)(1 − R2) c3βnpN∗ n µp(µ3+γ3)(1 − R3) · · · R2 np 1− Rn βnpN∗ n µp 0 0 0 · · · 0 0              . It is easy to see that rank KS·(I−K+KS)−1= 1 since all the columns of KS·(I−K+KS)−1are scalar multiples of its first column. Hence, KS·(I− K+KS)−1only has a nonzero eigenvalue, which implies ρ(KS·(I−K+KS)−1) = R2 1p 1− R1 +R2 2p 1− R2 +· · · +R2 np 1− Rn because the trace of a matrix is equal to the sum of its eigenvalues. ut In cases when the control strategy focuses on reducing the shedding rate of infectious hosts, we have the following result. Theorem 4 Consider the epidemiological model (12) and define the target set as U={(n+ 1,1),(n+ 1,2),...,(n+ 1, n)}. Provided that the condition max{R1,R2,...,Rn}<1 (21) is satisfied, the target reproduction number TUis given by TU=R2 1p 1− R1 +R2 2p 1− R2 +· · · +R2 np 1− Rn .(22) Proof It is straightforward to see that ρ(K−KU)<1⇐⇒ max{R1,R2,...,Rn}<1. Moreover, the matrix KU·(I−K+KU)−1is a lower triangular matrix with 0’s in the first nelements of its diagonal and R2 1p 1− R1 +R2 2p 1− R2 +· · · +R2 np 1− Rn as the last element of its diagonal. ut One of the advantages of using the target reproduction number is making a big problem smaller. Instead of struggling with the complete and probably very complex R0, one can handle a much smaller amount of information: some of the entries of the next-generation matrix. Therefore, although TSand TUhave the same value. In real life, one of the two strategies may have advantages over the other one. For instance, assuming Control Strategies in Multigroup Models 19 that we want to reduce the value of the entries Uof the next-generation matrix K(16) by a proportion more than 1 −1/TU, then we have to reduce the pathogen shedding rate or increase the recovery rate in each host class. But, if the objective is to reduce the entries Sin K; increasing the pathogen clearance rate µpreduces all of these entries at the same time. This could be a practical advantage of the last strategy over the first one. How multiple transmission routes affect the spread of many environmentallydriven diseases is not well-understood (Tien and Earn,2010). Nevertheless, the host-to-host transmission pathway has been traditionally considered as the predominant cause of disease spread (Bani-Yaghoub et al.,2012). For model (12), infectious individuals can transmit the infection both through direct contact with a susceptible person and also by pathogen shedding into the environment. One of the basic aims of this work is to explore the role of the host-environment-host indirect transmission pathway. Therefore, control strategies (A),(B) and (C) essentially focus on controlling environmental factors. 3.2 A metapopulation model with a central patch In this section, we consider a metapopulation model with n+1 patches that can be thought of as cities, countries, or other geographically autonomous regions with the particularity that one of them is the gravity center that connects the other ones. We propose a compartmental SIRS model for each patch. For the central patch, we are going to denote the total population at time tby Nc(t), whereas Sc(t), Ic(t) and Rc(t) give the number of susceptible, infectious and recovered individuals at time t, respectively. For the remaining npatches the classes are denoted by Si(t), Ii(t) and Ri(t) with i∈ {1,2, . . . , n}. So the total population at time tin patch iis Ni(t) = Si(t) + Ii(t) + Ri(t) for i∈ {1,2, . . . , n, c}:= J. Concerning movement, we assume that the members of the region i∈ {1,2, . . . , n}make short-term visits to the central region and return to the home patch, while individuals in the central patch can travel to all the remaining patches and then get back home. When infectious individuals are visiting another region they can infect the susceptible individuals of that region. Likewise, the susceptible ones that travel can be infected by infectious ones of the visited region. For the sake of clarity, the incidence will be governed by the mass action law and we also assume that all individuals are born susceptibles. The metapopulation SIRS model takes the following form ˙ Si=Λi−(βiiIi+βicIc)Si−µiSi+νiRi, ˙ Ii=(βiiIi+βicIc)Si−(µi+γi)Ii, ˙ Ri=γiIi−(νi+µi)Ri, (23) 20 Fernando Salda˜na, Ignacio Barradas for patches i∈ {1,2, . . . , n}. While for the central patch, ˙ Sc=Λc−βccScIc− n X i=1 βciScIi−µcSc+νcRc, ˙ Ic=βccScIc+ n X i=1 βciScIi−(µc+γc)Ic, ˙ Rc=γcIc−(νc+µc)Rc. (24) Here, Λiare recruitment rates, βij are the transmission rates of infectious individuals from patch jto susceptible individuals in patch i. The mortality rates are denoted by µiand γirepresent recovery rates. The average duration of the immune period for individuals in patch i∈Jis 1/νi. The graph associated to the dynamics of the infection process is shown in Fig. 4. For a metapopulation epidemic model patches should be connected through migration. This movement of individuals from one area to another can be short-term or long-term. The long-term movement appears when people move to another area and not necessarily come back to the home patch. Arino and Portet (Arino and Portet,2015) studied long-term movement in a metapopulation model whose infection process follows the same structure than system (23)-(24). Besides the type of movement, the difference between system (23)- (24) and Arino’s model is that we assumed a SIRS structure while in (Arino and Portet,2015) the authors omitted the possible loss of immunity of the individuals. To see that solutions of the model (23)-(24) with nonnegative initial conditions remain nonnegative consider when at least one of the phase space variables xiis equal to zero. Direct computation gives that if xi= 0, then ˙xi≥0 where x∈ {S, I, R}and i∈J. Thus, solutions cannot become negative for future times. The total population size of the complete system N=Pi∈JNi satisfies ˙ N≤Λ−µN, with Λ=X i∈J Λi, µ = min i∈J{µi}, therefore lim sup t→∞ N(t)≤Λ/µ. It follows that solutions of (23)-(24) are bounded. Moreover, for all the patches, the total population size of the patch converges to N∗ i=Λi/µiwith i∈J. Then, the region Γ={(S1, I1, R1, . . . , Sc, Ic, Rc)∈R3n+3 +:Si+Ii+Ri≤Λi/µi, i ∈J} is positively invariant under the flow of (23)-(24). The disease-free equilibrium ˜ Eobtained by solving for Siand Riafter setting Ii= 0, takes the values S∗ i=N∗ iand R∗ i= 0 for i∈J. Thus, ˜ E= (N∗ 1,0,0, . . . , N∗ n,0,0, N∗ c,0,0) (25) Control Strategies in Multigroup Models 21 Fig. 4: Topological structure of the infection process for model (23)-(24). The reproduction numbers Ricorrespond to the isolated dynamics of patch i∈ J, while the reproduction numbers Ric are associated with the commuting between the central patch and patch i, 1 ≤i≤n. is the unique disease-free equilibrium of model (23)-(24). As in the proof of Theorem 1, constructing a suitable Lyapunov function we can prove the following result. Theorem 5 Let Kbe the next-generation matrix for model (23)-(24)and R0=ρ(K). If the basic reproduction number satisfies R0≤1, then the disease-free equilibrium ˜ Eof the metapopulation model (23)-(24)is globally asymptotically stable in Γ. Since it has been proved the importance of the mobility component in the persistence of diseases in metapopulation models (Arino and Van den Driessche,2003) our control strategy shall focus on the interactions between patches rather than the local dynamics of each patch. Theorem 6 Provided that the condition max{R1,...,Rn,Rc}<1 (26) is fulfilled, the target reproduction number TZfor system (23)-(24) where the target set is defined as Z={(1, n + 1),(2, n + 1),...,(n, n + 1),(n+ 1,1),(n+ 1,2),...,(n+ 1, n)}is given by TZ=s1 (1 − Rc)R2 1c 1− R1 +R2 2c 1− R2 +· · · +R2 nc 1− Rn,(27) 22 Fernando Salda˜na, Ignacio Barradas where Ri=βiiN∗ i µi+γi , i ∈J and Ric =sβicβciN∗ iN∗ c (µi+γi)(µc+γc), i ∈ {1,2, . . . , n} are the basic reproduction numbers of the within-patch transmission and the infectious caused by the commuting between the i∈ {1,2, . . . , n}patches and the central patch, respectively (see Fig. 4). The proof follows the same idea that Theorem 3.1 and it is therefore omitted. Observe that in the calculated target reproduction number (27), in the denominator arise expressions of the form 1−Ri,i∈J. Thus, in a limit sense, the value of TZgoes to infinity as Ritend to 1. This seems to contradict the fact that TZ<1⇔ R0<1,and TZ= 1 ⇔ R0= 1 (28) when the target reproduction number is well defined. However, this is not a discrepancy since Ri,i∈J, are local reproduction numbers associated to submodels and relation (28) only applies for the basic reproduction number of the complete model. We also want to emphasize that the conceptual idea of the design of control strategies from the topological structure of the infection process is applicable to a broad range of models beyond those where the heterogeneity is spatial. For example, infectious process related to vector-borne diseases that affect multiple host species have an analogous structure that the one presented in Fig. 4, with the difference that the reproduction numbers associated to the within-species transmission are zero. In the above case, the vector has the function of transmitting the infection to the different species so it works as the common source of infection. 4 Discussion In recent years, the fast development of the field of mathematical epidemiology has included a wide range of multigroup mathematical models that consider host heterogeneities. However, much remains to be understood about how such heterogeneities and the topological structure of the infection process affect the spread of infectious diseases and our capacity to control them. Even though R0is a linear measure of the strength of the infection, under the homogeneous mixing assumption, it aptly determines the control effort needed to eradicate an infection (Roberts and Heesterbeek,2003) which is one of the fundamental tasks in the mathematical modeling of infectious diseases. However, for a large number of diseases there are multiple factors such as physiological, geographical or even economic conditions that affect the trend of the infection and therefore the homogeneous mixing assumption is no longer valid. This might make it necessary to consider different host types where each group guarantees homogeneous mixing. Once having several host types, the Control Strategies in Multigroup Models 23 basic reproduction number often appears as complex combinations of parameters that make any analysis difficult. In these cases, R0does not improve our understanding of how efforts should be focused to eliminate the epidemic (Olmos et al.,2015). In this paper, we bring forward some techniques to reduce the value of R0 for multigroup models even without explicitly knowing its expression. Given the original model, we studied some subsystems and computed their basic reproduction numbers. These basic reproduction numbers allowed us to express the target reproduction number in a more treatable way for concrete public health interventions. In other words, our methodology focuses on preventing outbreaks at the global level by means of controlling the local dynamics of the infection process. We started by applying these methods to a two-species model for leptospirosis. Three control policies were studied, all of them focusing on the indirect transmission caused by the interaction between hosts and the freeliving leptospira in the environment. In particular, by comparing the strategy that minimizes exposures to environmental risk factors with the one that reduces the pathogen shedding rate, we found that both target reproduction numbers are equal. Biologically, this means that both processes lead to the same outcome. Therefore, one can select the strategy that is most feasible according to the local conditions. The combination of the above strategies brings a stronger strategy because for endemic cases 1 <TW<TS=TUand therefore 1 −1/TW<1−1/TS. Yet, none of these strategies is good enough to control the infection when the direct transmission cycle is endemic. In order to study infections caused by pathogens with the ability to survive in the environment, we proposed a generalization of the previous model, namely the n-groups compartmental model (12). The model includes two potential routes of transmission: host-to-host and environment-to-host transmission. Traditionally, control strategies have been focused on the host-to-host transmission pathway. Nevertheless, the relative importance of the indirect infections caused via contact with a contaminated environment is uncertain (Holt et al.,2006;Bani-Yaghoub et al.,2012;Tien and Earn,2010). Therefore, we analyzed control strategies that center on the environment-to-host and computed the target reproduction number in each case to quantify the control effort needed to eradicate the infection. We also studied the spread of an infection among discrete geographical regions with a core region. Even though the connection of the regions is by way of short-term movement, it still allows infectious individuals to transmit the pathogen to susceptible individuals of a different region. Theorem 6implies that if the disease can persist even in one patch we cannot eradicate the epidemic by just controlling the movement between patches in the metapopulation model (23)-(24). In fact, we also notice that the infection can persist in the population even if all the basic reproduction numbers for the transmission within patches are less than one. Another important observation is that the topological structure of the infection process of model (12) is a particular case of the infection cycle that 24 Fernando Salda˜na, Ignacio Barradas follows the metapopulation model (23)-(24). The difference is that the common source of infection in model (12) (namely the free-living pathogen) does not have the ability to produce within-group infections. On the other hand, the central group of the metapopulation model can produce within-group infections with a reproduction number Rc. Also note that in qualitative terms, the strategy that aims to reduce the pathogen shedding by infectious hosts and minimize exposure to environmental risk factors in model (12), is equivalent to the strategy that focuses on reducing the infections caused by travel between patches in the metapopulation model. This explains the similarity in the target reproduction numbers (15) and (27), and lead us to remark that there is a connection among the cycles of infection, the basic reproduction number and the target reproduction number. These results illustrate that understanding the topological structure of the infection process and partitioning it in simple cycles is useful to design and evaluate control strategies. The focus of this work has been compartmental SIRS models composed of nhost groups with a common source of infection that fits the dynamics of a broad range of infectious diseases. 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