key: cord-0260550-wkjozgbg authors: Tocto-Erazo, M. R.; Olmos-Liceaga, D.; Montoya-Laos, J. A. title: Effect of daily human movement on some characteristics of dengue dynamics date: 2020-09-23 journal: nan DOI: 10.1101/2020.09.20.20198093 sha: a60edd67d12394aded174d096d4b39c1b6934ee6 doc_id: 260550 cord_uid: wkjozgbg Human movement is a key factor in infectious diseases spread such as dengue. Here, we explore a mathematical modeling approach based on a system of ordinary differential equations to study the effect of human movement on characteristics of dengue dynamics such as the existence of endemic equilibria, and the start, duration, and amplitude of the outbreak. The model considers that every day is divided into two periods: high-activity and low-activity. Periodic human movement between patches occurs in discrete times. Based on numerical simulations, we show unexpected scenarios such as the disease extinction in regions where the local basic reproductive number is greater than 1. In the same way, we obtain scenarios where outbreaks appear despite the fact that the local basic reproductive numbers in these regions are less than 1 and the outbreak size depends on the length of high-activity and low-activity periods. This work is divided in the following sections. The formulation of the model and the analysis 23 of uncoupled patches is given in Section 2. Then, in Section 3, we study the effect of daily human The classic vector-host mathematical model is given by the following system 2 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint where S, I and R represent the susceptible, infected and recovered population, respectively, and 1 P y Q the susceptible and infected mosquito population, respectively. 2 We include the daily periodic movement between two patches in model (1) as follows. The 3 interval [t k , t k+1 ) is the time period corresponding to the kth day and T l ∈ (0, 1) the fraction of the 4 day of low-activity such that for interval [t k , t k + T l ) we have in each patch only resident population 5 composed of N i individuals (i = 1, 2). Thus, the time interval [t k , t k + T l ) is named the low-activity 6 period. For a fixed day k, α i represents the proportion of the population from patch i that moves 7 every day to another patch j at time t k + T l and returns to patch i at time t k+1 . Thus, human 8 movement takes place on the time interval [t k + T l , t k+1 ) which is named the high-activity period. 9 For the low-activity period, the susceptible, infected and recovered human population from 10 patch i are represented by S l i , I l i and R l i , respectively, and the susceptible. The susceptible and 11 infected vector population from patch i are represented by P i and Q i , respectively. On the other 12 hand, for high-activity period, human population from patch i is divided into two subpopulations. The first subpopulation is composed of people from patch i who do not move to another patch, that 14 is, (1 − α i )N i . This subpopulation is subdivided into susceptible (S h ii ), infected (I h ii ) and recovered 15 (R h ii ). The second subpopulation is composed of residents from patch j who move to patch i, α j N j . 16 This subpopulation is subdivided into susceptible (S h ji ), infected (I h ji ) and recovered (R h ji ). Since 17 we assume that the vector population does not move between patches, susceptible and infected 18 vectors remain represented by P i and Q i , respectively. Thus, the following equations represent the 19 dynamics of the populations for the low-activity period [t k , t k + T l ): where N il := N i and i = 1, 2. For the high-activity period [t k + T l , t k+1 ), the set of equations become: where N ih := (1 − α i )N i + α j N j , and i, j = 1, 2, i = j. All model parameters are defined in Table 1 . 3 We observe that model (2)-(3) can be reduced to uncoupled patches in the form of system (2). This is done by taking T l = 1, that is, having only low-activity periods. 5 Parameter Meaning α i Proportion of humans from patch i who move to patch j at time t k + T la . N i Resident humans of patch i. In order to study our coupled model (2)-(3), we first make an analysis for each system separately 6 without considering a piecewise definition in time. Then, we focus on understanding the dynamics 7 of the daily human movement. . Then, the disease-free equilibrium 11 of system (2) is given by (S i ,Ī i ,R i ,P i ,Q i ) = (N il , 0, 0, Λ vi /µ vi , 0) and, using to the next generation is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint matrix approach as in [22] , the basic reproductive number (R il ) of uncoupled system is given by Previous work [21] has shown that if R il > 1, then there exists an endemic equilibrium In addition, authors in [21, 23] also have shown that the disease-free equilibrium is globally 4 asymptotically stable (GAS) when R il < 1, and the endemic equilibrium is GAS when R il > 1. For the high-activity period (3), we define S i * : Thus, the dynamics of uncoupled system (3) can be 7 written as: for each i = 1, 2. Since the structure of system (6) is the same as (2), results concerning the stability of the 10 equilibrium points are analogous to system (2). In particular, the disease-free and endemic 11 equilibrium points are given by (N il , 0, 0, Λ vi /µ vi , 0) and (S i * ,Ĩ i * ,R i * ,P i * ,Q i * )), respectively, where In addition, the basic reproductive number (R ih ) for uncoupled system (6) is given by 5 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . We observe that each R il and R ih does not depend on human movement, in this sense, 1 the theoretical results on the existence and stability of the equilibrium points are given for the 2 uncoupled system. However, when we the patches are coupled, these basic reproductive numbers 3 loses meaning and only provide information when the patches are uncoupled. In this case, a global 4 R 0 is not well defined so that we want to give an understanding of what occurs in each patch due 5 to the movement depending on the local basic reproductive numbers. In this section, we focus on understanding some effects due to daily human movement on the In general, the basic reproductive number (R 0 ) and the endemic equilibrium (I * ) of model 20 (2)-(3) for a disconnected patch with human population N can be written as and 22 From (9) and (10), we have that , and I * reaches 23 its maximum at pointN given by where a = δ + µ h . From Figure 1 , we observe that a patch with N smaller (larger) than N leads to 25 have R 0 > 1 (R 0 < 1). The basic reproductive number is a measure that gives conditions for the 26 existence of endemic equilibria and disease propagation in each patch separately. Thus, R 0 < 1 27 means that there is no favorable conditions for the disease spread, whereas R 0 > 1 implies that the 28 6 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . conditions are favorable for an outbreak to occur in each disconnected patch. In addition, while 1 N N . The findings in the previous subsection can be applied to see how the disease propagation 5 conditions change in each patch when there is human migration between them. For this, we define 6 A as the net population that move between patches, that is, A := |α 1 N 1l − α 2 N 2l |. Table 2 Table 2 are based on the value of N which is the 9 threshold population that generates or not endemic equilibria. The first column of the table shows 10 the value of the basic reproductive number in each patch before migration is considered (R 1l and 11 R 2l ). The second column shows the possible outcomes after a proportion of humans from patch 1 12 moves to patch 2, and vice versa (R 1h and R 2h ). The third column displays the conditions that the 13 populations must satisfy in order for every scenario to occur. The scenarios are used to understand 14 the daily human movement between patches. 15 7 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint Scenarios before Conditions migration migration In order to show how the displacement of people from one patch to another may influence 1 the disease propagation conditions, we examine the scenario R 1l < 1 and R 2l > 1, i.e, during the 2 low-activity period, in patch 1, the disease propagation conditions are not favorable, and in patch 2, 3 the conditions are favorable. To this, we consider the following resident populations: N 1l = 90000 4 and N 2l = 45000 for patch 1 and 2, respectively, and parameter values given in Table 3 . Based 5 on the parameter values, we obtain that R 1l = 0.68 and R 2l = 1.37. Figure 2 shows under which 6 conditions R 1h and R 2h are smaller or greater than 1, where the latter results in the existence of 7 endemic equilibria according to α 1 and α 2 values. Note that Figure 2 shows only the first three 8 outcomes for the case R 1l < 1 and R 2l > 1 given by Table 2 . Observe that there are no values of 9 α 1 and α 2 where both R 1h and R 2h are simultaneously greater than 1. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint From now on, to study the effect of daily periodic movement with complete model (2)-(3), we 1 take variables I 1 and I 2 to represent infected residents from patches 1 and 2, respectively. That is, for i, j = 1, 2, i = j. Observe that I i contabilize the infected individuals from patch i, no matter 3 where the disease was acquired. In this subsection, we study, by means of numerical simulations, some effects of daily human 6 movement on characteristics of the coupled model solutions, such as the existence of endemic 7 equilibria, and the start, duration, and amplitude of the outbreak. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint is preserved. Here we study the following cases of the presented scenario in Figure is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint patch 2 moves to patch 1. In the extreme case T l = 0.9 (the low-activity period is very 1 large), there is an endemic equilibrium in patch 2 due to the fact that almost all the time 2 the population remains in their residence patch and the basic reproductive number (R 2l ) 3 is greater than 1. In this case, we could approximate the R 0 value of patch 2 by the R 0 4 value of the disconnected patches. For individuals residing in patch 1, we observe that by 5 taking 10% of individuals from patch 1 who move to patch 2, for a short time period, it 6 is sufficient to generate endemic levels in patch 1 despite theoretically R 1l and R 1h are less 7 than 1. Now, for the extreme case T l = 0.1 (short low-activity period), the presence of an 8 endemic level in patch 1 is due to the fact that most of the time the 10% of the population 9 that belongs to patch 1, is in patch 2. This 10% carries the endemic levels aquired in patch . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint Cases α 1 = 0.5 and α 1 = 0.9 1 A similar behavior of existence and non-existence of endemic equilibria arise for these values 2 of α 1 and scenario R 1l < 1 and R 2l > 1. Regions of disease extinction can be more complex as is observed in Figure 5 , which shows is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. In this subsection, we present scenarios to observe some effects of the periodic human movement 2 on the outbreak dynamics. The parameter values from Table 3 are used for the numerical simulations. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . To end our study cases, we present scenarios where delay and advance of outbreaks are 2 observed when patches separately have conditions for the existence of outbreaks. 3 We first take N 1l = 50000, N 2l = 15000, α 1 = 0.5 and α 2 = 0.1. For these values, we obtain 4 R 1l = 1.23, R 2l = 4.11, R 1h = 2.33 and R 2h = 1.60. In Figure 9 , we notice that if the patches 5 are uncoupled (black dashed lines), the dynamics of both patches are governed by the R 1l 6 and R 2l values. In this case, the maximum incidence of cases in patch 2 is greater than in 7 patch 1, which coincides with the fact that R 2l is much larger than R 1l . Compared to the 8 dynamics of the decoupled patches, the outbreaks for T l = 0.98 occur earlier in patch 1, and 9 the one in patch 2 remains practically the same. In this case, the temporal dynamics of the 10 uncoupled system are inherited, that is, although the behavior of the outbreak in patch 1 11 is preserved, this outbreak is advanced because patch 2 has a high incidence of cases. As 12 the high-activity period increases, the maximum incidence in patch 1 goes up and the one 13 in patch 2 decreases. In addition, the outbreak in patch 2 is delayed as T l goes from 0.98 14 to 0.02. Clearly, these effects are due to R 1l < R 2l for the low-activity period, but during 15 the high-activity period, the intensity of the basic reproductive numbers is inverted, that is, is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . in Subsection 3.2. For this, we assume, without loss of generality, R 1h < R 2h . From Figure 10 11, we have that while R 1h decreases, R 2h take very large values. In fact, R 1h ∈ [0.68, 1.37], 11 while R 2h can be greater than 20. That is, while R 2h take values very high and R 1h is at 12 least 0.68, the outbreaks appear earlier and the maximum incidence of cases increase. . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . 1 In this work, our goal was to investigate how the daily human movement affects some characteristics This work studies the effects of daily commuters on the disease dynamics under a little-explored 10 approach, different from what is traditionally applied to multi-patch models. We believe that 11 modeling the disease spread where the division of more than one region is clearly defined, needs to is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint work is that it is possible to have a better biological description of the phenomenon. 1 From Figures 3 to 6 , the region of disease extinction varies greatly. We have observed that 2 these regions become larger when the basic reproductive numbers of the uncoupled patches are 3 relatively close to 1. Thus, this fact is dependent on the size of interacting populations and the 4 time spent by the populations in their residence patch. These results are not intuitive and might 5 have not been observed unless the movement between two patches is considered. 6 Other different scenarios might occur if we assume that the patches have different propagation 7 intensity not related to humans. For example, regions with different sanitary measures or with 8 abundant vegetation could lead to different rates of transmission from humans to mosquitoes or 9 vice versa, mosquito mortality rate, and mosquito recruitment rate. In fact, Table 2 14 Our approach can be useful not only for vector-borne diseases such as zika or chikungunya 15 but also for those with direct transmission such as SARS and COVID-19, diseases which might 16 generate pandemics due to human movement. In this respect, an infected human might be 17 exposed to different populations during its complete period of infection, leading to a more complex 18 understanding of the basic reproductive number and the disease dynamics. Moreover, to obtain a 19 generalization of the basic reproductive number for our complete model might be useful to establish 20 control policies that consider the human movement. Finally, although the study was mostly computational, it was quite complex. This is due to the 22 number of parameters involved in the model dynamics such as the population sizes of both patches, 23 the proportion of people moving between patches and the time period that individuals spend in 24 their residence patch. Therefore, it is not easy to have a complete theoretical understanding of 25 a system of this nature; however, it was useful to know some properties of the uncoupled model is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 23, 2020. . https://doi.org/10.1101/2020.09.20.20198093 doi: medRxiv preprint The global distribution and burden 4 of dengue Dengue and severe dengue Refining the Global Spatial Limits of Dengue 9 Virus Transmission by Evidence-Based Consensus The Role of Human Movement in the Transmission of Vector-Borne 13 House-to-house human movement drives dengue virus transmission Travel-associated dengue infections in the United States Man bites mosquito: Understanding the contribution of human 22 Non-linear Dynamics of Two-Patch Model Incorporating Secondary 1 Dengue Infection The interplay of vaccination and vector control on small 4 dengue networks Transmission dynamics for vector-borne diseases in a patchy environment Van Den Driessche, A multi-city epidemic model Mathematical Study of Dengue Disease Transmission in 11 The role of residence times in two-patch dengue transmission 14 dynamics and optimal strategies Vector-borne diseases models with residence times -A 17 Lagrangian perspective Assessment of optimal strategies in a two-patch dengue 20 transmission model with seasonality Assessing the effects of daily commuting in two-patch dengue 23 dynamics: A case study of Cali, Colombia Epidemic dynamics of a vector-borne 26 disease on a villages-and-city star network with commuters Analysis of a dengue disease transmission model