Decoding the double trouble A mathematical modelling of co-infection dynamics of SARS-CoV-2 and influenza-like illness

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Contents

1 Introduction 3

2 Model formulation and description 7

3 Qualitative Analysis 9

3.1 Infectious model with Covid-19 only . . . . . . . . . . . . . . . . 9

3.1.1 Invariant Region . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.2 Positivity of solution . . . . . . . . . . . . . . . . . . . . . 10

3.1.3 Disease free equilibrium (DFE) and basic reproduction

number R0C.......................... 10

3.1.4 Local Stability of DFE . . . . . . . . . . . . . . . . . . . . 10

3.1.5 Global Stability of DFE . . . . . . . . . . . . . . . . . . . 11

3.1.6 Endemic equilibrium analysis . . . . . . . . . . . . . . . . 11

4 Co-infection model system 12

5 Stability Analysis 13

5.0.1 Global Stability of DFE . . . . . . . . . . . . . . . . . . . 14

6 Invasion reproduction number R0Inv 15

7 Numerical Simulation 16

7.1 DataFitting.............................. 16

7.2 Real time R0Estimation....................... 19

7.3 Impact of transmission parameters . . . . . . . . . . . . . . . . . 21

7.4 Impactofcurerate.......................... 23

7.5 Timeseries .............................. 23

7.6 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Discussion and Conclusion 27

arXiv:2210.05649v1 [q-bio.PE] 11 Oct 2022

Decoding the double trouble: A mathematical

modelling of co-infection dynamics of SARS-CoV-2 and

inﬂuenza-like illness

Suman Bhowmicka,∗, Igor M. Sokolova,c, Hartmut H. K. Lentzb

aInstitute for Physics, Humboldt-University of Berlin, Newtonstraße 15, 12489 Berlin,

Germany

bFriedrich-Loeﬄer-Institut, Federal Research Institute for Animal Health, Institute of

Epidemiology, S¨udufer 10, 17493 Greifswald, Germany

cIRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany

Abstract

After the detection of coronavirus disease 2019 (Covid-19), caused by the se-

vere acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in Wuhan, Hubei

Province, China in late December, the cases of Covid-19 have spiralled out

around the globe. Due to the clinical similarity of Covid-19 with other ﬂu-

like syndromes, patients are assayed for other pathogens of inﬂuenza like ill-

ness. There have been reported cases of co-infection amongst patients with

Covid-19. Bacteria for example Streptococcus pneumoniae, Staphylococcus au-

reus, Klebsiella pneumoniae, Mycoplasma pneumoniae, Chlamydia pneumo-

nia, Legionella pneumophila etc and viruses such as inﬂuenza, coronavirus, rhi-

novirus/enterovirus, parainﬂuenza, metapneumovirus, inﬂuenza B virus etc are

identiﬁed as co-pathogens.

In our current eﬀort, we develop and analysed a compartmental based Or-

dinary Diﬀerential Equation (ODE) type mathematical model to understand

the co-infection dynamics of Covid-19 and other inﬂuenza type illness. In this

work we have incorporated the saturated treatment rate to take account of the

impact of limited treatment resources to control the possible Covid-19 cases.

As results, we formulate the basic reproduction number of the model system.

∗Corresponding author

Preprint submitted to Journal of L

X Templates October 12, 2022

Finally, we have performed numerical simulations of the co-infection model to

examine the solutions in diﬀerent zones of parameter space.

Keywords: Covid-19, Co-infection, ODE, Sensitivity Analysis, Invasion

Reproductive Number

1. Introduction

Coronavirus belongs to a group of enveloped virus with a single-stranded

RNA and viral particles bear a resemblance to a crown from which the name

originates. It belongs to the order of Nidovirales, family of Coronaviridae, and

subfamily of Orthocoronavirinae [1]. It can infect the mammals, including hu-

mans, giving rise to mild infectious disorders, sporadically causing to severe

outbreaks clusters, such as those brought about by the “Severe Acute Respira-

tory Syndrome” (SARS) virus in 2003 in mainland China [2].

After the ﬁrst reported case of new coronavirus disease outbreak in Wuhan,

Hubei province, People’s Republic of China, the virus has progressively dis-

seminated to diﬀerent countries in the world [3]. WHO declared it as a global

pandemic on 11 March, 2020 [4]. This disease can spread from person-to-person

through the breathing in of respiratory droplets from an infected person or

having the direct contact with contaminated surfaces [5].

The ending season and the culminating severity of the current Covid-19

pandemic wave are still unknown and unsettled issue. Meanwhile, the inﬂuenza

season has collided with the current pandemic that could pave the way for

more challenges. This possesses a larger threat to public health domain. It is

still uncertain how the seasonal inﬂuenza-like illness (ILI) will have an impact

on the long-term eﬀects on the course of Covid-19 pandemic. Both viruses

share similarities in transmission characteristics and alike clinical symptoms. ILI

and Covid-19 have been reported to cause respiratory infection. The interplay

between ILI and Covid-19 have been a major concern [6]. An emerging study

from England has shown that fatality amongst the people infected with both ILI

and Covid-19 are twice as that of someone infected with the new coronavirus

only [7]. An investigation conducted by the Public Health England (PHE)

has shown that people infected with the two viruses were having higher risk

of severe illness during the period from January to April 2020. According to

the same analysis, most cases of co-infection occurred in older people, and the

mortality rate was high. Reports from the USA point out that co-infection

between Covid-19 and other respiratory pathogens are notably more common

compared to the initial data what suggest such an interplay is rare found in

China [8, 9]. The authors in [10] report the co-infections between Covid-19

and other respiratory pathogens at a large urban medical centre in Chicago,

Illinois. A study [11] reveals that 7% of SARS-CoV-2-positive patients share

the burden of co-infection with other respiratory viruses. According to that

study the detection of other respiratory viruses in patients during this pandemic

assumes the Covid-19 co-infection. The authors in [12] note that high prevalence

of inﬂuenza-Covid-19 co-infection in their study. In [13], the authors describe

the cases of inﬂuenza and Covid-19 co-infection. During the last winter and

autumn, the co-circulation of inﬂuenza-Covid-19 has taken a toll on the health

of the patients and taxing the intensive care capacity [14, 15]. The authors

in the study [16] mention that co-infection with inﬂuenza A virus enhances

the infectivity of Covid-19. This is undoubtedly, a signiﬁcant threat that co-

infection of ILI and Covid-19 possess.

Proper and appropriate nursing and treatment processes can signiﬁcantly

reduce the outcome of epidemics in society. In traditional epidemiological mod-

elling assumption, the treatment rate is hypothesised to be constant or corre-

spond to the number of infected people and the recovery rate reckons on the

available medical resources such as test kits, ventilators, nursing facilities, eﬃ-

ciency of treatment etc. In the course of the ongoing pandemic, we have noticed

how this pandemic has stretched the healthcare systems of diﬀerent countries

around the globe while registering high mortality [17, 18, 19]. In the classi-

cal epidemiological models, it is very common to use the treatment function as

T(I) = ξI, I ≥0, where ξis positive but in the situation of sudden epidemic or

pandemic when the infected population is very large then it is not always pos-

sible to provide such a type of treatment which is proportional to the infected

number of individuals and Iis the number of infected people. To circumvent

the crux, the authors in [20] have introduced a constant treatment rate of the

form and demonstrated diﬀerent bifurcations. But to sustain such a constant

treatment rate might be plausible when there are a small number of infected

individuals as the medical resources are limited [21]. Following this, the authors

in [22, 23, 24] have modiﬁed the treatment rate by taking account of Holling

type II functional response as given below: T(I) = ηI

1+ζI and have explored the

model dynamics to understand the importance of limited medical resources and

facilities in the dissemination of infectious diseases. It is important to note that

this function is clearly an increasing function of Iand is bounded above by the

least upper bound η/ζ. This functional form of saturated treatment can provide

a better rationale for diﬀerent disease outbreaks such as SARS, Dengue etc[22]

in a new region because we know from our ongoing Covid-19 pandemic experi-

ence that in the beginning of an outbreak there is a lack of eﬀective treatment

due to either negligence or lack of knowledge about the disease. But afterwards

the treatment is being increased with the gain of knowledge about the disease

as well scaling up medical facilities [25]. Eventually, the treatment rate is out-

stretched to its maximum given the boundedness of medical resources of any

country [26]. Here, ηrepresents cure rate and ζdenotes the extent of the eﬀect

of the infected individuals being delayed for treatment [23].

Epidemiological models are quite beneﬁcial to examine the co-infection dy-

namics and to estimate the treatment facilities. There are several studies con-

centrating on the coexistence of two infectious agents in the susceptible hosts

[27, 28, 29, 30]. In [31], the authors have proved a suﬃcient condition for co-

existence of two infectious diseases. In the same vein, the authors in [32]have

proposed a new mathematical model Zika-dengue model while describing cou-

pled dynamics of Zika-dengue. The authors in [33] have asserted that pandemic

outbreaks can possibly be controlled by co-infection with other acute respiratory

infections that enhances the transmissibility of inﬂuenza virus. These suggest

that co-infection can potentially change the course of current ongoing pandemic

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摘要：

Contents1Introduction32Modelformulationanddescription73QualitativeAnalysis93.1InfectiousmodelwithCovid-19only................93.1.1InvariantRegion.......................93.1.2Positivityofsolution.....................103.1.3Diseasefreeequilibrium(DFE)andbasicreproductionnumberR0C........................

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