Abstract

The network-level reproduction number and extinction threshold for vector-borne diseases

Ling Xue , Caterina Scoglio

Kansas State Epicenter, Department of Electrical & Computer Engineering, Kansas State University,  U.S.  66506

E-mail: lxue@ksu.edu

Abstract

The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction contribute crucial knowledge on disease control, elimination, and mitigation of infectious diseases. Relationships between the basic reproduction numbers of two network-based ordinary differential equation vector-host models, and extinction thresholds of corresponding continuous-time Markov chain models are derived under some assumptions. Numerical simulation results for malaria and Rift Valley fever transmission on heterogeneous networks are in agreement with analytical results without any assumptions, reinforcing the relationships may always exist and proposing a mathematical problem of proving their existences in general. Moreover, numerical simulations show that the reproduction number is not monotonically increasing or decreasing with the extinction threshold. Key parameters in predicting uncertainty of extinction thresholds are identified using Latin Hypercube Sampling/Partial Rank Correlation Coefficient. Consistent trends of extinction probability observed through numerical simulations provide novel insights into mitigation strategies to increase the disease extinction probability. Research findings may improve understandings of thresholds for disease persistence in order to control vector-borne diseases.

Keywords: Extinction thresholds; Reproduction number; Network; Branching process; Vector-borne diseases

1 Introduction

Vector-borne diseases greatly impact health of humans and animals and are among the leading causes of worldwide death every year [16]; almost half of the world’s population is infected with at least one type of vector-borne diseases and millions of people die of vector-borne diseases every year [9]. These diseases also cause significant economic losses in areas of animal trade, agriculture, health care, tourism, as well as destroy ecosystems of throughout the world. Therefore, control and prevention of vector-borne diseases are both economical and humane. Efficient interventions require a good understanding of disease transmission and persistence, and dynamic modeling of vector-borne diseases may contribute greatly to this end [14]. A model may be used to learn many characteristics of an outbreak such as: whether or not an outbreak may occur, the size of the outbreak, the duration time of the outbreak, or the probability for the epidemic to die out [6]. Efficient mitigation strategies deduced from model results may stop an outbreak at early stages by reducing spreading parameters [6].

Globalization of trade and travel is one of the key factors driving the emergence of vector-borne diseases; heterogeneous structure also plays an important role on dynamics of infectious diseases [20]. Modeling the spatial spread of vector-borne diseases is a challenging task [3], but one possible approach is to consider a meta-population as a directed graph, or a network, with each vertex representing a subpopulation in a location, and links placed between two locations if there is a possibility of transmission, such as movement or proximity [5]. Network models are more widely used in epidemiology to understand the spread of infectious diseases through connected populations [27, 39].

The basic reproduction number, defined as the number of secondary cases produced by an infected individual in a naive population [12] is an important threshold on epidemiology among many others, such as type reproduction number [32], target reproduction [34], and threshold index for epidemicity [18]. The basic reproduction number is an important metric, predicting whether a disease will spread or die out in deterministic population and communicable disease theory [2]. If , one infectious individual generally produces more than one infection, leading to spread of an epidemic, whereas if , one infectious individual generates less than one infection on average [10], and epidemic may die out [12]. The same trajectory can always be observed with deterministic models given the same initial conditions [21]. If it is possible for an epidemic to occur again, a real world epidemic does not allow us to observe that the same infection happens to the same person at the same time [21]. Moreover, deterministic models have the shortcoming that the number of infected individuals may go to less than one [23].

In comparison, Markov chain models are more realistic in the sense of only taking integer values instead of continuously varying quantities [23] and taking into account chances by approximating or mimicking the random or probabilistic factors. The last infectious individual may recover before the infection is transmitted to other susceptible individuals so that the disease may become extinct [23]. Consequently, an infection introduced to a completely susceptible population may not invade the system even if [23]. Threshold for the extinction of an infectious disease to occur and probability of disease extinction are of interests. Bienaymé-Galton-Watson branching processes are widely used to study extinction of diseases involving multi-type infections.

Lloyd [23] reviewed theory of branching processes and computed extinction probability using branching processes for Ross malaria model [33] taking into account stochasticity and heterogeneity. Pénisson [28] presented several statistical tools to study extinction of populations consisted of different types of individuals, and their behaviors before extinction and in the case of a very late extinction. Allen and Lahodny Jr [1] computed reproduction numbers for deterministic models, and extinction thresholds for corresponding continuous-time Markov chain (CTMC) models using continuous-time branching process, and derived their relationships. A CTMC model is a stochastic counterpart of a deterministic ordinary differential equation (ODE) model [1]. Lahodny Jr and Allen [22] estimated probability of disease extinction for a Susceptible-Infected-Susceptible (SIS) multipatch model and illustrated some differences between thresholds for deterministic models and stochastic models numerically. Allen and van den Driessche [2] established connections between extinction thresholds for continuous-time models and discrete-time models and illustrated the relations through numerical simulations. Although probability of disease extinction is defined as the probability for the number of infections to become zero when time goes to infinity, various numerical approximations for many types of models within finite time showed good agreement with predicted extinction probability using branching processes [1, 2, 22].

Deriving relationships between reproduction numbers and extinction thresholds is a complex task for vector-borne diseases transmitted on heterogeneous networks due to too many parameters and large size matrices. According to current knowledge, very little research has studied it. The objectives of this research are to relate the extinction threshold, in a stochastic setting and the reproduction number, in a deterministic setting for vector-host meta-population models, as well as gain understandings in how to increase extinction probability.

The contribution of our work is summarized as follows.

  1. Relationships between extinction thresholds and the reproduction numbers are derived for network-based vector-host models under some assumptions.

  2. Numerical simulations show that the relationships still exist after removing above assumptions.

  3. Consistent trends of extinction probability varying with disease parameters are observed through extensive numerical simulations.

  4. The key parameters in predicting uncertainty of the extinction threshold are identified using Latin Hypercube Sampling/Partial Rank Correlation Coefficient (LHS/PRCC).

  5. The relationship between varying disease parameters and potential mitigation strategies is biologically interpreted.

This paper is organized as follows. Section 2 reviews the next generation matrix approach for computing and the branching process for deriving . Section 3 calculates for a deterministic vector-host model in which transmission dynamics of vectors are described by a Susceptible-Infected (SI) model and transmission dynamics of hosts are described by an SIS model. We relate of corresponding CTMC model and analytically. In Section 4, an analogue of results in Section 3 is obtained for a model in which transmission dynamics of vectors are described by a Susceptible-Exposed-Infected (SEI) model and transmission dynamics of hosts are described by a Susceptible-Exposed-Infected-Recovered (SEIR) model. Local transmission and trans-location transmission due to proximity for vector-borne diseases are both considered in Sections 3 and 4. In Section 5, the relationships derived in Sections 3 and 4 are numerically shown to hold without any assumptions for simplified malaria and Rift Valley fever meta-population models. The sensitivity test sorted out the key parameters in predicting uncertainty of extinction probability. Relationships between varying parameters and extinction probabilities are explored through extensive simulations for homogeneous populations and a two-node network. Section 6 provides a summary and discussion of mathematical derivations and simulation results.

2 Preliminary

The next generation matrix approach used to compute for compartmental models is reviewed here, followed by a review of the multitype branching process approximation used to derive for corresponding CTMC models.

2.1 Computation of using the next generation matrix approach

The next generation matrix approach is frequently used to compute . In this section, we quickly review this approach. For more details, we refer to [11, Chapter 5], [38]. For simplicity, let stand for compartments that are only related to infected and asymptomatically infected individuals. The original nonlinear system of ODEs including these compartments can be written as , where and represent new infections and transfer between compartments, respectively. Moreover, represents the rate at which new infections appear in compartment , and , where (resp. ) represents the rate at which individuals transfer from (resp. into) compartment . The Jacobian matrices representing transmission, and representing transition are defined as:

(1)

where denotes disease free equilibrium (DFE), and is the number or proportion of infected individuals in compartment , where . Matrix is nonnegative and is a nonsingular M-matrix.

Matrix is called the next generation matrix. The entry of indicates the expected number of new infections in compartment produced by the infected individual originally introduced into compartment , where .

The reproduction number, , is defined as the spectral radius of , denoted by .

2.2 Deriving using branching process approximation

Calculating the probability of disease extinction is one of the most interesting applications of branching process. The branching process may lead to disease extinction or persistence. We are interested in the conditions under which a disease may become extinct and the probability for this event to occur. First, we review the approach of using branching process to compute extinction threshold and extinction probability for multi-type infections.

We refer to [1, 28] for the rest of this section. Let be a set of discrete-valued vector random variables. Assume that individuals of type produce individuals of type and that the number of infected individuals produced by type are independent of the number of infected individuals produced by other individuals of type or type for . Additionally, individuals of type have identical probability generating function (pgf). Let be the offspring random variables for type , where is the number of infected individuals of type produced by individuals of type . The probability that one individual of type produces infected individuals of type is given as

The offspring pgf array , is defined as

(2)

Note that a trivial fixed point of always exists at .

We denote by the expectation matrix of offspring distribution which is nonnegative, where represents the expected number of new infected individuals of type produced by an individual of type .

The extinction threshold, is defined as the spectral radius of the expectation matrix, denoted by .

Recall that and assumptions in [28] are as follows.

  1. is not simple. Here, a function is called simple if it is linear with no constant term.

  2. Matrix is irreducible.

If , under assumptions and , the pgf has at most one fixed point in , denoted by , if extinction array in exists. In the following, extinction array only refer to . If , then disease extinction probability, denoted by , is

(3)

If , then

3 SI vector model and SIS host metapopulation model

In this section, a deterministic vector-host model in which disease transmission dynamics of vectors are described by an SI model, while transmission dynamics of hosts are described by an SIS model. The reproduction number and extinction threshold for corresponding CTMC model are analytically related.

3.1 The reproduction number

The model for vectors consists of compartment representing susceptible vectors, and compartment representing infected vectors. Disease dynamics of hosts are modeled by an SIS model.

(4)

The recruitment rate of vectors (resp. hosts) in node is (resp. ) for all . The rate of new infections in vectors in node produced by local hosts, and hosts in other nodes are and , respectively. The death rate of susceptible and infected vectors in node are and , respectively. The rate of host infection in node produced by local vectors, and vectors in other nodes are and , respectively. The death rates of susceptible and infected hosts in node are and , respectively. The rate of recovery for hosts in node is .

Since and , are only compartments related to infected and asymptomatically infected, system of ODEs can be rewritten as follows.

A unique solution at DFE, represented by exists, where and . The Jacobian matrices and defined in (1) for this model are

where

(5)
(6)

Here

The notation represents the diagonal matrix with diagonal entries . To calculate , we first prove the following lemma.

Lemma 1.

Let be square matrices of the same size and , then

Proof.

For any ,

(7)

Therefore, if .

If , we assume that . Then there exists a such that . By (7), for a nonzero , contradicting the assumption that . Therefore,

A direct calculation gives By Lemma 1, we have the following proposition.

Proposition 1.

The reproduction number of the model (4) is

(8)

3.2 The threshold for extinction probability

In this section, we compute for corresponding CTMC of model . See Table 1 for state transitions and rates.

The pgfs are:

where , the index for , represents , and represents .

Description State transition Rate
Host birth
Death of
Host local infection
Host infection by
Host recovery
Death of
Vector birth
Death of
Vector local infection
Vector infection by
Death of
Table 1: State transitions and rates for corresponding continuous-time Markov chain for deterministic model omitting node index .

The expectation matrix is:

(9)

where are the same as those in (5), and

Note that if both and are positive matrices, then the assumptions and in [28] hold for this model.

Lemma 2.

Let be nonnegative square matrices with the same size such that is an eigenvalue of and be nonnegative diagonal matrices such that for some real numbers . Then the spectral radius of satisfies that

Proof.

Since by Theorem in [40],

(10)

By hypothesis and (7), is an eigenvalue of . Following the fact that for any if ,

Similarly, Lemma follows (10) and Lemma 1. ∎

Remark 1.

If both and are positive matrices, then is an eigenvalue of by Perron-Frobenius theorem.

By Lemma 2, we have the following proposition.

Proposition 2.

The extinction threshold of model (4) satisfies that

3.3 The relationship between and

To obtain a theoretical relationship between in and , we assume that

(11)

for constant numbers throughout this section. The assumption can be interpreted biologically as: the probability of natural death is identical for vectors from each node, and the probability of natural death is identical for hosts from each node. The assumption shall be removed for numerical simulations in the next section.

Theorem 1.

Under the assumption (11),

  • If or , then ;

  • If or , then .

Proof.

Under the assumption (11), , , and , where is the identity matrix. Therefore, in (9) can be rewritten as follows,

Without loss of generality, we assume that . By Lemma 2 and (8),

(12)

Following (12),

Theorem follows the above two inequalities. ∎

Corollary 1.

If the further assumption is made that except assumption (11), then if and only if . Moreover, .

Proof.

By Theorem 1 (1), if , then . Assuming that , by Theorem 1 (2), , which is a contradiction. Conversely, if , then following a similar argument. Hence, if and only if . This proves the first part. The second part directly follows Theorem 1. ∎

4 SEI vector model and SEIR host metapopulation model

A deterministic model in which vectors are divided into compartments , and , and hosts are classified into compartments , and is presented. The reproduction number for this model and the extinction threshold for corresponding CTMC model are connected.

4.1 The reproduction number

The following model extends the model in Section 3.1 by adding compartment for exposed vectors, and compartment for exposed hosts. Other terms have identical meanings as corresponding ones in model . The rate at which exposed vectors and exposed hosts in node transfer to infected compartments are and , respectively.

(13)

Compartments related to infected and asymptomatically infected are and , . The unique solution at DFE is , where and are the same as those in Section 3.1. The above system of ODEs including these compartments can be rewritten as follows.

The Jacobian matrices and at DFE are

where and are given in ; matrices and are in Equation ; and

By a direct calculation,

Following Lemma 1,

Proposition 3.

The reproduction number of the model (13) is

(14)

4.2 The threshold for extinction probability

State transitions and rates for corresponding CTMC of model are listed in Table 2.

Description State transition Rate
Host birth
Death of
Death of
Death of
Death of
Host local infection
Host infection by
Host recovery
Host Latent to infectious
Vector birth
Death of
Death of
Death of
Vector local infection
Vector infection by
Vector Latent to infectious
Table 2: State transitions and rates for corresponding continuous-time Markov chain for deterministic model omitting node index .

The pgfs are:

where represents only , represents , represents , and represents